^12 PHILOSOPHICAL TRANSACTIONS. [anNO 1705. 



expressed by all the different combinations of such indices only (being integer 

 numbers) whereof no one may exceed the highest index of the things exposed, 

 and being more than one in a combination, are each of them, which are in the 

 same combination, comprehended in a distinct index thereof; these expressions 

 of n will necessarily be the several forms of the combinations taken n and n, 

 whereof m things are capable ; whence is derived a general theorem for finding 

 the combinations and alternations of m things taken n and n universally : i. e. 

 whether m consist of things all different or not, and whether n be equal to, or 

 less than m. 



Theorem. — If n be expressed, according to all the different forms of com- 

 bination which the things exposed are capable of, 



!p = the highest index r a = the number of p' '\ 

 q = the next highest ^ (3 = the number of 9* f in every form of 

 r = the next highest \ y = the number of r* ^ combination ; 

 s ^ the next highest (. «J = the number of s* J 

 &c. 



!a = the number of all the indices not less than p ^ 

 B = the number of all the indices not less than q f which are in the things 

 c = the number of all the indices not less than r ^ exposed ; 

 D = the number of all the indices not less than s j 

 &c. 

 and A = «-|-p, c = ^ + y, d = c -\- i, &c. 



Then the number of the combinations of m things taken n and n, in any one 

 form of combination, shall be ^ x a- 1 x a~2 ^ , - « x b - « - 1 ^^ 



X ^-^ — X c — — ^^^ V — c X p-c— ^^^ continued to so many terms 



yxv — 1 d' X ^ — 1 ^ 



as there are different indices in the form of combination, and each term to «, |3, 

 y, ^, &c. places respectively ; and this number multiplied into 



nx»--l X« — 2x« — 3xn— 4x« — 5xw— 6 &c. continued to n placet 

 l>xp— 1 xp — 2x &c.|» X 9 X 9 — 1 X &c.|^ X r X r — 1 X &c.|>' X &c. each »erie» 



continued to jb, ^, r, &c. places respectively, shall be the number of their 

 alternations. 



But the sum of all the combinations and alternations which are in every form 

 of n, shall be the whole number of combinations and alternations of m things 

 taken n and n. 



Demonstration, — 1st, Then it is evident, that those combinations, which are 

 in different forms, differ from each other. Again, it is evident that the com- 

 binations of m things, as a^ b^ cP dJ^ e«/« g« A'' z% &c. (the indices simply con- 

 sidered) taken n and n, in a form wherein are />', q', and r*, shall be equal to the 

 number of the combinations of the />•, which are in the things exposed, taken 



