30 BOOK OP MATHEMATICAL PROBLEMS. 



157. In the expansion of (a l + a a +... +a p ) n , where n is a 

 whole number not greater than p, prove that the coefficient of any 

 term in which none of the quantities a l , a a ... a p appears more 

 than once as a factor is [n. 



158. The number of permutations of n different letters taken 

 all together, in which no letter occupies the same place as in a 

 certain given permutation, is 



159. Prove that 



nr = 9 + f r ~ 2 n + ^r-1 + 3 + /?,-* -... + (- 1)' (?+ 1). 



160. The number of combinations of 2n things taken n toge- 

 ther, when n of the things and no more are alike, is 2" j and the 

 number of combinations of Zn things, n together, when n of the 

 things and no more are alike, is 



161. The number of ways in which mn different things can 

 be distributed among m persons so that each person shall have n 



\mn 



of them is rr^. 

 1 1*) 



162. There are p suits of cards, each suit consisting of q 

 cards numbered from 1 to q\ prove that the number of sets of q 

 cards numbered from 1 to q which can be made from all the suits 

 is p 9 . 



1G3. The number of ways in -which p things may be dis- 

 tributed among q persons, so that everybody may have one at 

 least, is 



