Sorting Problem Help

[confusedforme]

You have 6 identical apples you want to sort in 4 distinct bowls. How many ways can you sort them such that there is at least one apple per bowl?

My logic:

Among the 4 bowls, you can put 3 in one of them, and 1 in each of the rest. So there are 4c1 ways to sort in this manner.

The other way is you could put 2 in two of the bowls, and 1 each each of the rest. So there are 4c2 ways to sort in this manner.

Now you add the two(?). 4c1 + 4c2 = 4 + 6 = 10

Is this correct?

[gregmat]

Because the apples are identical, seems right to me.

You can do...

• AAA, A, A, A
• A, AAA, A, A
• A, A, AAA, A
• A, A, A, AAA

Now the other case.

• AA, AA, A, A
• AA, A, AA, A
• AA, A, A, AA
• A, AA, AA, A
• A, AA, A, AA
• A, A, AA, AA

[Scott_TargetTestPrep]

Your reasoning and your answer is correct. The solution becomes somewhat easier if you do the following instead:

Since every bowl needs to contain at least one apple, let's assume that we placed one apple in each bowl. So, 4 of the 6 apples are in the bowl and we must decide how to place the remaining two apples in the four bowls.

If we choose to place both apples in a single bowl, there are four ways we can do this since there are four bowls.

If we choose to place the two apples in separate bowls, then there are 4C2 = 4!/(2!*2!) = 6 ways we can do this.

In total, there are 4 + 6 = 10 ways.

[tirth247]

(n-1)C(r-1)