Quandaries and Queries


In how many ways can 12 different fruits be distributed among 3 boys so that each receives at least 1 fruit?

I solved the problem as follows-
No. of selection of 3 fruits from 12 fruits=12C3
No. of ways 3 fruits can be distributed among 3 boys so that each receives 1 fruit=3!
Of the 9 remaining fruits each can be distributed in 3 ways.So,all 9 fuits can be distibuted in 3^9 ways.
Hence,the answer=12C3x3!x39.

The answer in my book is different.I want to know the error in my procedure.




I bet that the answer in your book is smaller!

12 different fruits:
A = apple, B = banana, C = cranberry, ..., K = kiwi, L = lemon.

3 boys:
X = Xavier, Y = Yiang, Z = Zachary

- One of the 12C3 ways to pick three fruits from the 12 is to pick {A, B,

- And one of the 3! ways to distribute one these among the three boys so
each receives a fruit is X gets A, Y gets B, Z gets C.

- And one of the 39 ways to distribute the remaining fruits is
X gets {D, E}, Y gets {F, G, H} and Z gets {I, J, K, L}.

So the final distributlon is {A, D, E} to X, {B, F, G, H} to Y
and {C, I, J, K, L} to Z.

But there is at least one other way to arrive at that distribution:
- The initial pick of three fruits among twelve could have been { E, G, K
the distribution of these could have been E to X, G to Y and K to Z,
and then the distribution of the remaining fruits could have been
{A, D} to X, {B, F, H} to Y and {C, I, J, L} to Z.

Therefore, your count of 12C3x3!x39 counts that specific distribution
more than once, this is why it is not the right answer.

Actually, the distribution {A, D, E} to X, {B, F, G, H} to Y
and {C, I, J, K, L} to Z is counted exactly 3x4x5 = 60 times
by your method; can you see why? This means that you would
be on the right track if all you needed to do was to divide your
answer by 60. Unfortunately, not all the distributions are counted
exactly 60 times by your method; for instance the distribution
{A, B} to X, {C, D} to Y and {E, F, G, H, I, J, K, L} to Z
is counted only 2x2x8 = 32 times. This means that you cannot
get the right answer by dividing your answer by a number;
the best is to start over with a different method.

P.S. I don't really know names of fruits starting with all
the missing letters D, E, F, G, H, I, J.



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