Problem Solving - 21

Mrs. Napueta has been given film vouchers. Each voucher allows the holder to see a film without charge. She decides to distribute them among her four nephews so that each nephew gets at least two vouchers. How many vouchers has Mrs. Napueta been given if there are 120 ways that she could distribute the vouchers?

(A) 13

(B) 14

(C) 15

(D) 16

(E) more than 16


Answer: C

Formula - Number of ways of distributing n identical things among r persons when each person may get any number of things = C(n+r-1, r-1)

Hence

Let total number of vouchers = n

All 4 nephews get atleast 2 vouchers, thus remaining vouchers to be distributed among 4 nephews where each can get 0 or more vouchers = n-8

Using the formula above we get = C(n-8+4-1, 3) = (n-8+4-1)C3 = 120 = (n-5)(n-6)(n-7) = 120 * 3! = 720

Hence n = 15