ELECTIONS, or CHOICE

ELECTIONS, or CHOICE, signify the several different ways of taking any number of things proposed, either separately, or as combined in pairs, in threes, in fours, &c.; not as to the order, but only as to the number and variety of them. Thus, of the things $a,\; b,\; c,\; d,\; e$, &c. the elections of one thing are $(a,\; )\; 1\; =\; 2^1\; -\; 1$, two things are $(a,\; b,\; ab,\; )\; 3\; =\; 2^2\; -\; 1$, three things are $(a,\; b,\; c,\; ab,\; ac,\; bc,\; abc,\; )\; 7\; =\; 2^3\; -\; 1$, &c.; and of any number $n$, all the elections are $2^n\; -\; 1$; that is, one less than the power of 2 whose exponent is $n$, the number of single things to be chosen, either separately or in combination.