In how many ways can 5 identical apples be put into 3 different bowls

Question

In how many ways can 5 identical apples be put into 3 different bowls such that at least 1 apple is included in each bowl?

A. 6
B. 10
C. 142
D. 150
E. 243

tgubbay1 · Answer

(*Disclaimer: not necessarily the right answer, just my attempt)

In order for all baskets to have at least 1 apple, you can have two types of distributions:
i) 3 - 1 - 1 (Can happen 3x different ways since the baskets are different)
ii) 2 - 2 -1 (Can also happen 3x different ways since the baskets are different)

I learned something through this exercise! Since the apples are identical - There will only be 1 way of choosing them. Therefore the first basket has 1 way of choosing 3 apples (not 5C3). Therefore the number of ways to choose apples so that each basket has at least 1 will be:

i) (1x1x1) x 3
+ 
ii) (1x1x1) x3 = 6 ways

Answer is A

Bunuel  - is this correct?

SN_09 · Answer

Hi  Bunuel ,

Can we please have the official explanation posted here? Just want to verify the method I have used to arrive at the answer.

Thank you!

Bunuel · Answer

An official solution isn't available, but here's how I would approach the problem:

Since each bowl must contain at least one apple, we can start by distributing 3 apples as {1 - 1 - 1}, leaving 2 apples to be allocated.

When distributing the 2 remaining apples one at a time, there are 3 possible ways: {0 - 1 - 1}, {1 - 0 - 1}, or {1 - 1 - 0}.

When placing both remaining apples to a single bowl, there are also 3 possible ways: {2 - 0 - 0}, {0 - 2 - 0}, or {0 - 0 - 2}.

Therefore, there are a total of 3 + 3 = 6 ways.

Answer: A.

SN_09 · Answer

Perfect, this is helpful. Thanks a ton!

bumpbot · Answer

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In how many ways can 5 identical apples be put into 3 different bowls

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In how many ways can 5 identical apples be put into 3 different bowls

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