Question

18. How many ways are there to distribute k balls into n distinct boxes (k < n) with at most one ball in any box if

(a) The balls are distinct?

(b) The balls are identical?

40. How many nonnegative integer solutions are there to the pair of equations x1 + x2 +···+ x6 = 20 and x1 + x2 + x3 = 7?

56. How many ways are there to distribute 20 toys to m children such that the first two children get the same number of toys if

(a) The toys are identical? 

(b) The toys are distinct?

58. How many ways are there to distributer identical balls into n distinct boxes with exactly m boxes empty?

Answer 1

18.

(a) if balls are distinct

then , number of ways = n!/(n-k)!

(b) if balls are identical

then, number of ways

=$\binom{n}{k}$

= n!/k!(n-k)!

Answer 2

40.

x1 + x2 + x3 = 7, C(7+2,2) = 36

x4 + x5 + x6 = 13, C(13+2,2) = 105

number of non-negative integer solutions = 36*105 = 3780