Ten apples are distributed among Amy, Ben, Corin, and Dora such that Amy gets at least two, and Corin and Dora get at least one each. If Ben gets at least as many apples as all the others put together, how many ways can the apples be distributed?
A ≥ 2 ... C ≥ 1 ... D ≥ 1 ... B ≥ A + C + D ... A + B + C + D = 10
list the possibilities ... there aren't that many ... I'll start
A2 , C1 , D1 , B6
A2 , C2 , D1 , B5
that's half
Thanks R_scott
To find the number of ways the apples can be distributed, we can break down the problem into cases.
Case 1: Amy gets 2 apples
If Amy gets 2 apples, Ben needs to get at least 2 apples to have more than Amy and Corin and Dora need to get at least 1 each. This leaves 4 apples for Ben to distribute among himself, Corin, and Dora. Since Ben needs to have at least twice as many apples as Corin and Dora combined, we can consider the possible distribution of apples among just Ben, Corin, and Dora in the ratio 1:1:2. This means Ben gets 2 apples, and Corin and Dora each get 1 apple. There is only one possible distribution for this case.
Case 2: Amy gets 3 apples
If Amy gets 3 apples, Ben still needs to get at least 2 apples to have more than Amy. This leaves 5 apples for Ben, Corin, and Dora to distribute among themselves. If we consider the possible distribution of apples among Ben, Corin, and Dora in the ratio 1:1:3, this means Ben gets 3 apples, and Corin and Dora each get 1 apple. There is only one possible distribution for this case as well.
Case 3: Amy gets 4 apples
If Amy gets 4 apples, Ben needs to get at least 3 apples to have more than Amy. This leaves 3 apples for Ben, Corin, and Dora to distribute among themselves. Again, considering the possible distribution of apples among them in the ratio 1:1:3, Ben gets 3 apples, and Corin and Dora each get 1 apple. There is only one possible distribution for this case.
In total, there is only one possible distribution for each case, which means there are 3 total ways the apples can be distributed.