A man is distributing his coin collection with 35 coins to h

A man is distributing his coin collection with 35 coins to his five grandchildren. How many ways are there to distribute the coins if:

a. The coins are all the same

b. The coins are all distinct

c. The coins are the same and each grandchild gets the same number of coins.

d. The coins are all distinct and each grandchild gets the same number of coins.

Solution

Let\'s call the amount-to-be-divvied N, where we know that 5 < N < 35. Let n_k denote the amount left over after k rounds of giving. We\'ll also say that n₀=N, since there are N coins left to be distributed after 0 rounds of giving.

Thus, we have

n₀ = N, while

n_k+1 = (5/35)(n_k-1). This expression is because the butler starts the (k+1)^st round by taking his coin, then the (k+1)^st child takes of what\'s left (there\'s n_k-1 left at this point),

Thus,

n₀ = N

n₁ = (5/35)(N-1)

n₂ = (5/35)[(5/35)(N-1)-1], etc.

I\'ll also let you verify that

n₅ = (5³⁵*N - sum[i=5 to 1] 4⁵*35^5-i )/5⁵, and that this simplifies to

n₅ = (1024*N - 8404)/5⁵. We\'re almost done, hang in there! What this tells us is that, since n₅ must be an integer (this is what\'s left for the grandchildren), we have

1024*N - 8404 = 0 (mod 5⁵). But, we also know that n₅ must be divisible by another factor of 5, since the 5 grandchildren split n₅ equally between themselves. Thus, all told, we have