COMBINATION

COMBINATION, in mathematics, is the variation or alteration of any number of quantities, letters, or the like, in all the different manners possible. See CHANGES.

Aphorisms. I. In all combinations, if from an arithmetic decreasing series, whose first term is the number out of which the combinations are to be formed, and

whose common difference is 1, there be taken as many terms as there are quantities to be combined; and these terms be multiplied into each other: and if from the series 1, 2, 3, 4, &c. there may be taken the same number of terms, and they be multiplied into each other, and the first product divided by the second; the quotient will be the number of combinations required. Therefore, if you would know how many ways four quantities can be combined in seven, multiply the first four terms of the series, 7, 6, 5, 4, &c. together, and divide the product, which will be 840, by the product of the first four terms of the series, 1, 2, 3, 4, &c. which is 24, and the quotient 35 will be the combinations of 4 in 7. II. In all permutations, if the series 1, 2, 3, 4, &c. be continued to as many terms as there are quantities to be changed, and those terms be multiplied into each other; the product will be the number of permutations sought. Thus, if you would know how many permutations can be formed with five quantities, multiply the terms 1, 2, 3, 4, 5 together, and the product 120 will be the number of all the permutations.

Problems. I. To find the number of changes that may be rung on 12 bells. It appears by the second aphorism, that nothing is more necessary here than to multiply the numbers from 1 to 12 continually into each other, in the following manner, and the last product will be the number sought.

II. Suppose the letters of the alphabet to be wrote so small that no one of them shall take up more space than the hundredth part of a square inch: to find how many square yards it would require to write all the permutations of the 24 letters in that size. By following the same method as in the last problem, the number of permutations of the 24 letters will be found to be 62,044,840,173,223,943,936,000. Now the inches in a square yard being 1296, that number multiplied by 100 gives 129600, which is the number of letters

Combination. letters each square yard will contain; therefore if we divide 62,044,840,175,323,943,936,000 by 129600 the quotient, which is 478,741,050,720,092,160, will be the number of yards required, to contain the above-mentioned number of permutations. But as all the 24 letters are contained in every permutation, it will require a space 24 times as large; that is, 11,489,785,217,282,211,840. Now the number of square yards contained on the surface of the whole earth is but 617,197,435,008,000, therefore it would require a surface 18620 times as large as that of the earth to write all the permutations of the 24 letters in the size above-mentioned.

III. To find how many different ways the eldest hand at piquet may take in his five cards. The eldest hand having 12 cards dealt him, there remain 20 cards, any five of which may be in those he takes in; consequently we are here to find how many ways five cards may be taken out of 20. Therefore, by aphorism I. if we multiply 20, 19, 18, 17, 16, into each other, which will make 1860480, and that number be divided by 1, 2, 3, 4, 5, multiplied into each other, which make 120, the quotient, which is 15504, will be the number of ways five cards may be taken out of 20. From hence it follows, that it is 15503 to 1, that the eldest hand does not take in any five certain cards.

IV. To find the number of deals a person may play at the game of whist, without ever holding the same cards twice. The number of cards played with at whist being 52, and the number dealt to each person being 13, it follows, that by taking the same method as in the last experiment, that is, by multiplying 52 by 51, 50, &c. so on to 41, which will make 3,954,242,643,911,239,680,000, and then dividing that sum by 1, 2, 3, &c. to 13, which will make 6,227,020,800, the quotient, which is 635,013,559,600 will be the number of different ways 13 cards may be taken out of 52, and consequently the number sought.

The construction of this table is very simple. The line A a consists of the first 12 numbers. The line A b consists every where of units; and second term 3, of the line B c, is composed of the two terms 1 and 2 in the preceding rank: the third term 6, in that line, is formed of the two terms 3 and 3 in the preceding rank: and so of the rest; every term, after the first, being composed of the two next terms in the preceding rank: and by the same method it may be continued to any number of ranks. To find by this table how often any number of things can be combined in another number, under 13, as suppose 5 cards out of 8; in the eighth rank look for the fifth term, which is 56, and that is the number required.

Though we have shewn in the foregoing problems the manner of finding the combination of all numbers whatever, yet as this table answers the same purpose, for small numbers, by inspection only, it will be found useful on many occasions; as will appear by the following examples.

V. To find how many different sounds may be produced by striking on a harpichord two or more of the seven natural notes at the same time. 1. The combinations of two in seven, by the foregoing triangle are - - - - - 21
2. The combinations of 3 in 7, are - - - - - 35
3. The combinations of 4 in 7, are - - - - - 35
4. The combinations of 5, are - - - - - 21
5. The combinations of 6, are - - - - - 7
6. The seven notes all together once - - - - - 1

Therefore the number of all the sounds will be - - - - - 120

VI. Take four square pieces of pasteboard, of the same dimension, and divide them diagonally, that is by drawing a line from two opposite angles, as in the figures, into 8 triangles; paint 7 of these triangles with the primitive colours, red, orange, yellow, green, blue, indigo, violet, and let the eighth be white. To find how many chequers or regular four-sided figures, different either in form or colour, may be made out of those eight triangles. First, by combining two of these triangles, there may be formed either the triangular square A, or the inclined square B called a rhom. Secondly, by combining four of the triangles, the large square C, may be formed; or the long square D, called a parallelogram.

Now the first two squares, consisting of two parts out of 8, they may each of them, by the eighth rank of the triangle be taken 28 different ways, which makes

Combina-
tion. makes 56. And the last two squares, consisting of four parts, may each be taken by the same rank of the triangle 70 times, which makes 140 To which add the foregoing number 56

And the number of the different squares that may be formed of the 8 triangles, will be 196

VII. A man has 12 different sorts of flowers, and a large number of each sort. He is desirous of setting them in beds or flourishes, in his parterre. Six flowers in some, 7 in others, and 8 in others; so as to have the greatest variety possible; the flowers in no two beds to be the same. To find how many beds he must have. 1. The combinations of 6 in 12 by the last rank of the triangle, are 924
2. The combinations of 7 in 12, are 792
3. The combinations of 8 in 12, are 495

Therefore the number of beds must be 2211

VIII. To find the number of chances that may be thrown on two dice. As each die has 6 faces, and as every face of one die may be combined with all the faces of the other, it follows, that 6 multiplied by 6, that is 36, will be the number of all the chances; as is also evident from the following table:

It appears by this table, 1. That the number of chances for each point continually increases to the point of seven, and then continually decreases till 12: therefore if two points are proposed to be thrown, the equality, or the advantage of one over the other, is clearly visible*. 2. The whole number of chances on the dice being 252, if that number be divided by 36, the number of different throws on the dice, the quotient is 7: it follows therefore, that at every throw there is an equal chance of bringing seven points. 3. As there are 36 chances on the dice, and only 6 of them doublets, it is 5 to 1, at any one throw, against throwing a doublet.

By the same method the number of chances upon any number of dice may be found: for if 36 be multiplied by 6, that product, which is 216, will be the chances on 3 dice; and if that number be multiplied by 6, the product will be the chances on 4 dice, &c.

COMBINATIONS of the Cards. The following experiments, founded on the doctrine of combinations, may possibly amuse a number of our readers. The tables given are the basis of many experiments, as well on numbers, letters, and other subjects, as on the cards; but the effect produced by them with the

last is the most surprising, as that which should seem to prevent any collusion, that is, the shuffling of the cards, is on the contrary the cause from whence it proceeds.

It is a matter of indifference what numbers are made use of in forming these tables. We shall here confine ourselves to such as are applicable to the subsequent experiments. Any one may construct them in such manner as is agreeable to the purposes he intends they shall answer.

To make them, for example, correspond to the nine digits and a cipher, there must be ten cards, and at the top of nine of them must be wrote one of the digits, and on the tenth a cipher. These cards must be placed upon each other in the regular order, the number 1 being on the first, and the cipher at bottom. You then take the cards in your left hand, as is commonly done in shuffling, and taking off the two top cards, 1 and 2, you place the two following, 3 and 4, upon them; and under those four cards the three following 5, 6, and 7: at the top you put the cards 8 and 9, and at the bottom the card marked 0. Constantly placing in succession 2 at top and 3 at bottom, and they will then be in the following order:

8.9.3.4.1.2.5.6.7.0

If you shuffle them a second time, in the same manner, they will then stand in this order:

6.7.3.4.8.9.1.2.5.0

Thus, at every new shuffle, they will have a different order, as is expressed in the following lines:

1 shuffle 8.9.3.4.1.2.5.6.7.0

2 6.7.3.4.8.9.1.2.5.0

3 2.5.3.4.6.7.8.9.1.0

4 9.1.3.4.2.5.6.7.8.0

5 7.8.3.4.9.1.2.5.6.0

6 5.6.3.4.7.8.9.1.2.0

7 1.2.3.4.5.6.7.8.9.0

It is a remarkable property of this number, that the cards return to the order in which they were first placed, after a number of shuffles, which added to the number of columns that never change the order, is equal to the number of cards. Thus the number of shuffles is 7, and the number of columns in which the cards marked 3, 4, &c. never change their places is 3, which are equal to 10, the number of the cards. This property is not common to all numbers; the cards sometimes returning to the first order in less number, and sometimes in a greater number of shuffles than that of the cards.

Combination. These tables, and the following examples at piquet except the 36th, appear to have been composed by M. Guyot.

Order before dealing. After 1st deal. After the 2d. After the 3d.

Order before dealing. After 1st deal. After the 2d. After the 3d.

Order before dealing. After 1st deal. After the 2d. After the 3d.

I. "Several letters that contain no meaning, being "wrote upon cards, to make them, after they have "been twice shuffled, give an answer to a question "that shall be proposed; as for example, What is "love?" Let 24 letters be wrote on as many cards, which, after they have been twice shuffled, shall give the following answer.

A dream of joy that soon is o'er.

First, write one of the letters in that line on each of the cards*. Then write the answer on a paper, and assign one of the 24 first numbers to each card, in the following order:

A D R E A M O F J O Y T H A T S O O N

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

I S O ' E R.

20 21 22 23 24

Next, write on another paper a line of numbers, from 1 to 24, and looking in the table for 24 combinations you will see that the first number after the second shuffle is 21, therefore the card that has the first letter of the answer, which is A, must be placed against that number, in the line of numbers you have just made†. In like manner the number 22 being the second of the same column, indicates that the card which answers to the second letter D of the answer, must

Combina- must be placed against that number: and so of the rest.
tion. The cards will then stand in the following order:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
O O F S A M N T O I S R H A E O ' E J O
20 21 22 23 24.
R A D Y T

From whence it follows, that after these cards have been twice shuffled, they must infallibly stand in the order of the letters in the answer.

Observe 1. You should have several questions, with their answers, consisting of 24 letters, wrote on cards: these cards should be put in cases, and numbered, that you may know to which question each answer belongs. You then present the questions; and when any one of them is chose, you pull out the case that contains the answer, and shewing that the letters wrote on them make no sense, you then shuffle them, and the answer becomes obvious.

2. To make this experiment the more extraordinary, you may have three cards, on each of which an answer is wrote; one of which cards must be a little wider, and another a little longer, than the others. You give these three cards to any one, and when he has privately chose one of them he gives you the other two, which you put in your pocket without looking at them, having discovered by feeling which he has chose. You then pull out the case that contains the cards that answer to his question, and perform as before.

3. You may also contrive to have a long card at the bottom, after the second shuffle. The cards may be then cut several times, till you perceive by the touch that the long card is at bottom, and then give the answer; for the repeated cuttings, however often, will make no alteration in the order of the cards.

The second of these observations is applicable to some of the subsequent experiments, and the third may be practised in almost all experiments with the cards. You should take care to put up the cards as soon as the answer has been shewn: so that if any one should desire the experiment to be repeated, you may offer another question, and pull out those cards that contain the answer.

Though this experiment cannot fail of exciting at all times pleasure and surprize, yet it must be owned that a great part of the applause it receives arises from the address with which it is performed.

II. "The 24 letters of the alphabet being wrote upon so many cards, to shuffle them, and pronounce the letters shall then be in their natural order; but that not succeeding, to shuffle them a second time, and then shew them in proper order." Write the 24 letters on the cards in the following order:

1 2 3 4 5 6 7 8 9 10 11 12
R S H Q E F T P G U X C
13 14 15 16 17 18 19 20 21 22 23 24
N O D Y Z I K & A B L M

The cards being disposed in this manner, shew them upon the table, that it may appear they are promiscuously marked. Then shuffle and lay them again on the table, pronouncing that they will be then in alphabetical order. Appear to be surprised that you have failed; take them up again and give them a second shuffle, and then counting them down on the table they

will all be in their natural order.

III. "Several letters being wrote promiscuously upon 32 cards, after they have been once shuffled, to find in a part of them a question; and then shuffling the remainder a second time, to shew the answer. Suppose the question to be, What is each Briton's boast? and the answer, His liberty; which taken together contain 32 letters."

After you have wrote those letters on 32 cards, write on a paper the words his liberty, and annex to the letters the first ten numbers thus:

H I S L I B E R T Y
1 2 3 4 5 6 7 8 9 10

Then have recourse to the table of combinations for ten numbers, and apply the respective numbers to them in the same manner as in experiment I. taking the first column, as these are to be shuffled only once, according to that order.

1 2 3 4 5 6 7 8 9 10
I B S L E R T H I Y

This is the order in which these cards must stand after the whole number 32 has been once shuffled, so that after a second shuffle they may stand in their proper order. Next dispose the whole number of letters according to the first column for 32 letters: the last ten are to be here placed in the order above; as follows,

W H A T I S E A C H B R I T O N ' S
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
B O A S T?
18 19 20 21 22

I B S L E R T H I Y
23 24 25 26 27 28 29 30 31 32

Therefore, by the first column of the table, they will next stand thus:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
I T B R O N S C H B O A E A S T long card.
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
I I S B S L I B E R T W H H I Y

You must observe that the card here placed the 16^th in order, being the last of the question, is a long card; that you may cut them, or have them cut, after the first shuffle, at that part, and by that means separate them from the other ten cards that contain the answer.

Your cards being thus disposed, you shew that they make no meaning; then shuffle them once, and cutting them at the long card, you give the first part to any one, who reads the question, but can find no answer in the others, which you open before him; you then shuffle them a second time, and shew the answer as above.

IV. "To write 32 letters on so many cards, then shuffle and deal them by twos to two persons, in such manner, that the cards of one shall contain a question, and those of the other, an answer. Suppose the question to be, Is nothing certain? and the answer, Yes, disappointment."

Over the letters of this question and answer, write the following numbers, which correspond to the order in which the cards are to be dealt by two and two.

I S N O T H I N G C E R T A I N?
31 32 27 28 23 24 19 20 15 16 11 12 7 8 3 4
Y E S, D I S A P O I N T M E N T.
29 30 25 26 21 22 17 18 13 14 9 10 5 6 1 2

Then have recourse to the first column of the table for 32 numbers, and dispose these 32 cards in the following order, by that column.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
O I E R G C A N T P I N T A I S
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
T M E H S D I N N O Y N T E I S

The cards being thus disposed, shuffle them once, and deal them two and two; when one of the parties will necessarily have the question, and the other the answer.

Instead of letters you may write words upon the 32 cards, 16 of which may contain a question, and the remainder the answer; or what other matter you please. If there be found difficulty in accommodating the words to the number of cards, there may be two or more letters or syllables wrote upon one card.

V. "The five beatitudes." The five blessings we will suppose to be, 1. Science, 2. Courage, 3. Health, 4. Riches, and 5. Virtue. These are to be found upon cards that you deal, one by one, to five persons. First write the letters of these words successively, in the order they stand, and then add the numbers here annexed to them.

S C I E N C E C O U R A G E
31 26 21 16 11 6 1 32 27 22 17 12 7 2
H E A L T H R I C H E S
28 23 18 13 8 3 29 24 19 14 9 4
V I R T U E
30 25 20 15 10 5

Then range them in order agreeable to the first column of the table for 32 numbers, as in the last experiment. Thus,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
L H N A T E R E U A C R G T I U
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
E E C I I C H S O H R E E V S C

Next, take a pack of cards, and write on the four first the word Science; on the four next the word Courage and so of the rest.

Matters being thus prepared, you show that the cards on which the letters are wrote convey no meaning. Then take the pack on which the words are wrote, and spreading open the first four cards, with their backs upward, you desire the first person to choose one. Then close those cards and spread the next four to the second person; and so to all the five; telling them to hold up their cards lest you should have a confederate in the room.

You then shuffle the cards, and deal them one by one, in the common order, beginning with the person who chose the first card, and each one will find in his hand the same word as is wrote on his card. You will observe, that after the sixth round of dealing, there will be two cards left, which you give to the first and second persons, as their words contain a letter more than the others.

VI. "The cards of the game of piquet being mixed together, after shuffling them, to bring, by cutting them, all the cards of each suit together." The order in which the cards must be placed to produce the effect desired being established on the same principle as that explained in the experiment II. except that the shuf-

fling is here to be repeated three times, we think it will be sufficient to give the order in which they are to be placed before the first shuffle.

You then shuffle the cards, and cutting at the wide card, which will be the seven of hearts, you lay the eight cards that are cut, which will be the suit of hearts, down on the table. Then shuffling the remaining cards a second time, you cut at the second wide card, which will be the seven of spades, and lay, in like manner, the eight spades down on the table. You shuffle the cards a third time, and offering them to any one to cut, he will naturally cut them at the wide card*, which is the seven of diamonds, and consequently divide the remaining cards into two equal parts, one of which will be diamonds and the other clubs.

VII. "The cards at piquet being all mixed together, to divide the pack into two unequal parts, and name "the number of points contained in each part." You are first to agree that each king, queen, and knave shall count, as usual, 10, the ace 1, and the other cards according to the number of the points. Then dispose the cards, by the table for 32 numbers, in the following order, and observe that the last card of the first division must be a wide card.

Combination. You then shuffle them carefully, according to the method before described, and they will stand in the following order.

The cards being thus disposed, you ask your adversary in what suit you shall repique him. If he say in clubs or diamonds, you must deal the cards by threes, and the hands will be as follows,

When the cards are by shuffling disposed in this order, you cut them at the wide card, and pronounce that the cards you have cut off contain 66 points, and consequently the remaining part 194.

VIII. "The Inconceivable Repique." When you would perform this experiment with the cards used in the last, you must observe not to disorder the first 10 cards in laying them down on the table. Putting those cards together, in their proper order, therefore, you shuffle them a second time in the same manner, and offer them to any one to cut, observing carefully if he cut them at the wide card, which will be the ace of hearts, and will then be at top; if not, you must make him, under some pretence or other, cut them till it is; and the cards will then be ranged in such order that you will repique the person against whom you play, though you let him choose (even after he has cut) in what suit you shall make the repique.

Order of the cards after they have been shuffled and cut.

VOL. III.

If he against whom you play, who is supposed to be elder hand, has named clubs for the repique, and has taken in five cards, you must then lay out the queen, knave, and nine of diamonds, and you will have, with the three cards you take in, a fixiem major in clubs, and quatorze tens. If he leave one or two cards, you must discard all the diamonds.

If he require to be repiqued in diamonds, then discard the queen, knave, and nine of clubs; or all the clubs, if he leave two cards; and you will then have a hand of the same strength as before.

Note, If the adversary should discard five of his hearts, you will not repique him, as he will then have a septiem in spades: or if he only take one card: but neither of these any one can do, who has the least knowledge of the game. If the person against whom you play would be repiqued in hearts or spades, you must deal the cards by twos, and the game will stand thus:

12 M

Combination. If he require to be repiqued in hearts, you keep the quint to a king in hearts, and the ten of spades, and lay out which of the rest you please: then, even if he should leave two cards, you will have a fixiem major in hearts, and quatorze tens, which will make a repique.

But if he demand to be repiqued in spades; at the end of the deal you must dexterously pass the three cards that are at the bottom of the stock (that is, the ten of clubs, ten of diamonds, and ace of hearts) to the top*, and by that means you reserve the nine, king, and ace of spades for yourself: so that by keeping the quint in hearts, though you should be obliged to lay out four cards, you will have a fixiem to a king in spades, with which and the quint in hearts, you must make a repique.

Observe here likewise, that if the adversary lay out only three cards, you will not make the repique: but that he will never do, unless he be quite ignorant of the game, or has some knowledge of your intention.

This last stroke of piquet has gained great applause, when those that have publicly performed it have known how to conduct it dexterously. Many persons who understand the nature of combining the cards, have gone as far as the passing the three cards from the bottom of the stock, and have then been forced to confess their ignorance of the manner in which it was performed.

IX. "The Metamorphosed Cards." Provide 32 cards that are differently coloured; on which several different words are wrote, and different objects painted. These cards are to be dealt two and two, to four persons, and at three different times, shuffling them each time. After the first deal, every one's cards are to be of the same colour; after the second deal, they are all to have objects that are similar; and after the third, words that convey a sentiment.

Dispose of the cards in the following order.

The cards thus coloured, figured, and transcribed, are to be put in a case, in the order they here stand.

When you would perform this experiment you take the cards out of the case, and show, without changing the order in which they were put, that the colours, objects, and words are all placed promiscuously. You then shuffle them in the same manner as before, and deal them, two and two, to four persons, observing that they do not take up their cards till all are dealt, nor mix them together: and the eight cards dealt to each person will be found all of one colour. You then take each person's cards, and put those of the second person under those of the first, and those of the fourth person under those of the third. After which you shuffle them a second time, and having dealt them in the same manner, on the first person's cards will be painted all the birds; on the second person's cards, all the butterflies; on those of the third, the oranges; and on those of the fourth, the flowers. You take the cards a second time, and observing the same precautions, shuffle and deal them as before, and then the first person, who had the last time the birds in his hand, will have the words that compose this sentence.

Sing, dear birds; I long to hear your enchanting notes.

The second person, who the last deal had the butterflies, will now have these words:

Of an inconstant lover your changes present me the image.

The third, who had the oranges, will have this sentence:

As in my Phyllis, I find in you beauty and sweetness.

The fourth, who had the flowers, will have these words:

Charming flowers, adorn the bosom of my shepherdess.

It seems quite unnecessary to give any further detail, as they who understand the foregoing experiments will easily perform this.

Among the different purposes to which the doctrine of combinations may be applied, those of writing in cipher, and deciphering, hold a principal place. See the article CIPHER.