COMBINATION. 



of individuals, but afterwards became 

 much more extensive. The accuser first 

 swore to the truth of his accusation ; the 

 accused gave him the lie : upon which 

 each threw down a gage, or pledge of 

 battle, and the parties were committed 

 prisoners to the day of combat. See 

 CHAMPION. 



COMBINATION, in mathematics, is 

 the variation or alteration of any number 

 of quantities, letters, sounds, or the like, 

 in all the different manners possible. It 

 is shewn, in the memoirs of the French 

 Academy, that two square pieces, each 

 divided diagonally into two colours, may 

 be combined 64 different ways, so as to 

 form so many different kinds of chequer- 

 work ; which appears surprising enough, 

 when one considers that two letters or 

 figures can only be combined twice. See 

 CHANGES. 



COMBINATION, doctrine of. Prob. 1. 

 Any number of quantities being given, 

 together with the number in each com- 

 bination, to find the number of combina- 

 tions. One quantity admits of no combi- 

 nation : two, a and b, only of one combi- 

 nation ; of three quantities, a b c, there are 

 three combinations, viz. a b, a c, b c , of 

 four quantities, there are six combina- 

 tions, viz, a b,a c,a d, b c,b d, c d , of five 

 quantities, there are ten combinations, 

 viz. a b, a c, b c, a d, b d, c d y cte, b e,c e, 

 de. Hence it appears that the numbers 

 of combinations proceed as 1, 3, 6, 10 ; 

 that is, they are triangular numbers, 

 whose sides differ by unity from the 

 number of given quantities. If this then 

 be supposed q, the side of the number of 

 combinations will be q 1 and so the 



number of combinations - -? 



See TRIANGULAR NUMBERS. 



If three quantities are to be combined, 

 and the number in each combination be 

 three, there will be only one combina- 

 tion, a b c; if a fourth be added, four 

 combinations will be found, a b c, a b d, 

 b c d, a c d,- if a fifth be added, the com- 

 binations will be ten, viz. abc, a b d,b c d, 

 a c d, a b e, b d e, bee, ace, a d e ; if a 

 sixth, the combinations will be twenty, 

 &c. The numbers, therefore, of combi- 

 nations proceed as 1, 4, 10, 20, &c. that 

 is, they are the first pyramidal triangular 

 numbers, whose side differs by two units 

 from the number of given quantities. 

 Hence, if the number of given quantities 

 be q, the side will be q 2, and so the 



number of combinations ' ^ 



If four quantities are to be combined, 

 we shall find the numbers of combina- 

 tions to proceed as pyramidal, triangular 

 numbers of the second order, 1, 5. 15, 

 35, &c. whose side differs from the num- 

 ber of quantities by the exponent minus 

 an unit. Wherefore, if the number of 

 quantities be 9, the side will be 9 3, 



and the number of combinations - - J 



See PYRAMIDAL 



2 3 4 



NUMBERS. 



Hence is easily deduced a general 

 rule of determining the number of com- 

 binations in any case whatsoever. Sup- 

 pose, for example, the number of quan- 

 tities to be combined 9, and the expo- 

 nent of combination n; the number of 



.. . 9 n + 1 q n+2, 



combinations will be - - - ' 



g-^S.g-^-H, &c t . u the number 



to be added be equal to n. Take 9 = 6 

 and n = 4, the number of combinations 

 will b 6 4 +* 6 ~ 4 + 2 6 4 + 3 



6 4 + 4 6 3 6 2 6 1 6+0 

 4 ~ 1 2 " 3 4 



3456 



If it be required to know all the possi- 

 ble combinations of the given quantities, 

 beginning with the combinations of the 

 several two's, then proceeding to three's, 



&c. we must add~^- ^-~- '^r ^-77 



* ,. '' 2 ' 1 2 



9+0. 9 3 9 2 9 1 9 + 0. 

 __ _ _ _ _ 



Whence the number of all the possible 

 combinations will be rn + "^"o 



9~ 2 , gg-lg 2g 3 gg l 

 3 "*"! ^2 3 4 ' ' 1 2 



^-5 ? g which is the sum of 

 o 4 o 



the uncix of the binomial raised to the 

 power 9, and abridged of the exponent 

 of the power increased by unity q + 1. 

 Wherefore, since these uncise come out 

 1 + 1, by being raised to the power 9; 

 and since 1 + 1 is equal to 2, 29 9 1 

 will be the number of all the possible 

 combinations. For example, if the num- 

 ber of quantities be 5, the number of pos- 

 sible combinations will be 25 6=32 

 6=26. 



Prob. 2. Any number of quantities be- 

 ing given, to find the number of all these 

 changes which these quantities, combined 1 



