# The number of ways in which 6 pencils can be distributed between two boys such that each boy gets at least one pencil is

The number of ways in which 6 pencils can be distributed between two boys such that each boy gets at least one pencil is

I think it is 5, but the answer is 62. Where am I wrong?

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Looks as if the pencils have different colours. –  André Nicolas Dec 30 '13 at 5:08
@AndréNicolas, thanks. –  Silent Dec 30 '13 at 5:15
You are welcome. The problem should have specified whether the pencils are distinguishable or not. Without that, there is ambiguity, and your interpretation (and answer) are perfectly reasonable. –  André Nicolas Dec 30 '13 at 5:19

The answer is indeed $5$ if one considers the pencils to be indistinguishable.

But if each pencil is distinguishable (say, a different color) then there are $2^6 = 64$ ways to distribute the pencils, $2$ of which leave one boy with no pencils. Leaving a total of $64-2=62$ ways.

The logic behind $2^6$ is that each of the $6$ pencils has $2$ places it can go. So there are $6$ factors of $2$.

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Thank you so much, Sir! –  Silent Dec 30 '13 at 5:16
Nice solution Eric Thoma...... –  juantheron Dec 30 '13 at 14:36

in order to have atleast 1 pencil with each boy, give 1 pencil to each. Now remaining 4 can be distributed among two in 5 ways only so I think u r correct.(pencils are same)

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