AND ON COMBINATIONS AND PERMUTATIONS. 



479 



Answer for 2 ; — .3.4.5.6= 15. 



1.2.3.4 



1.2.3.4 



[9. 10 . 11 . 12 - 5 .6. 7 .8 .9 + 10. 3 .4.5 .6] = 15. 



6. In cases of Combination, such as those to which formulas (xiv.) and (xv.) apply, when it is 

 required to determine the number of Combinations corresponding, not merely to one or two powers 

 of ,r, but to the entire range of the values of u, from to [a -i- /3 + 7, Sec] = o- in the former case, 

 and from to sa = a in the latter, the expression (xi.) for the product of the s Polynomes suggests 

 the following method for determining arithmetically the entire series of the Coefficients. The 

 method will be best explained by an example. 



How many Combinations can be formed from the Six Elements A, BB, CCC, taking 0, 1, 2i 

 3, 4, 5, 6 of them at a time. 



I 



The law of the terms in the last line, which contains the answer, deserves notice : viz. that the 

 terms corresponding to the indices u and 6 — u, are equal. 



How many different Combinations can be formed from the Four Elements AA, BB, taking 

 0, 1, 2, 3, 4 at a time.' 



7. From the given numbers of tlie Combinations formed by r elements of t different kinds 

 which combine o at a time, and by ct - t elements, of 8 — t different kinds, which combine u - v 

 III a time, it is rc(|iiircd to determine the numbers of the Combinations formed by those elements 

 Vol. VIII. 1'aut IV. sQ 



