 #1
Crystal037
 161
 5
 Homework Statement:
 In how many ways can 5 prizes be distributed among 4 boys when everybody can take any number of prizes.
 Relevant Equations:

no. of prizes=5
no. of boys=4
the first boy has 6 ways,
either he wins 0 or 1 or 2 or 3 or 4 or 5 ways
According to the first boy, the second boy also has 0 or 1 or 2 or 3 or 4 or 5 chances
According to the first and second boy, the other boy also has 0 or 1 or 2 or 3 or 4 or 5 chances.
So, the answer should be 4(either of them gets all the prizes)+4C1*3C1(First gets 4 prizes and the second gets 1) +4C1*3C1(where First gets 3 prizes and the second gets 2)+4C2*2C1(2 of them gets 1 prize each and one gets only 3 prizes)+4C2*2C1(2 of them gets 2 prizes each and one gets only 1 prize)+4C1*3C3(where one gets 2 prizes and all the other get 1 each)
=4+12+12+12+12+4
=56
But the correct answer is 1024 ways.
either he wins 0 or 1 or 2 or 3 or 4 or 5 ways
According to the first boy, the second boy also has 0 or 1 or 2 or 3 or 4 or 5 chances
According to the first and second boy, the other boy also has 0 or 1 or 2 or 3 or 4 or 5 chances.
So, the answer should be 4(either of them gets all the prizes)+4C1*3C1(First gets 4 prizes and the second gets 1) +4C1*3C1(where First gets 3 prizes and the second gets 2)+4C2*2C1(2 of them gets 1 prize each and one gets only 3 prizes)+4C2*2C1(2 of them gets 2 prizes each and one gets only 1 prize)+4C1*3C3(where one gets 2 prizes and all the other get 1 each)
=4+12+12+12+12+4
=56
But the correct answer is 1024 ways.