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# If I buy 7 identical bags of candy for 3 friends, how many ways can I distribute the candy to my 3 friends so that each friend receives at l

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If I buy 7 identical bags of candy for 3 friends, how many ways can I distribute the candy to my 3 friends so that each friend receives at least 1 bag of candy?

If I buy 7 distinguishable postcards, how many ways can I send the postcards to my 3 friends so that each friend receives at least 1 postcard?

Jan 16, 2020
edited by mathmathj28  Jan 21, 2020
edited by mathmathj28  Jan 23, 2020

#1
+109739
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(a) While travelling abroad, I bought 7 identical bags of candy for 3 friends. How many ways can I distribute the candy to my 3 friends, so that each friend gets at least one bag of candy?

This is similar to  distributing  7 identical balls into  3 distinct boxes with the  restriction  that each  box must contain at least 1 ball

The number of ways is given by

C (7-1, 3 - 1)  =  C ( 6,2)   =  15 ways

Jan 16, 2020
edited by CPhill  Jan 16, 2020
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That makes sense, thank you so much for explaining it!!

mathmathj28  Jan 19, 2020
#2
+109739
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(b) I also bought 7 different postcards. How many ways can I send the postcards to my 3 friends, so that each friend gets at least one postcard?

Let k be the number of postcards   and n  be the number of friends

The  number of ways  =

S (k, n) * n!

Where  S(k, n)  is  a  Stirling Number of the Second Kind

So  we have

S(7,3) * 3!  =

301  *  6  =

1806 ways

Jan 16, 2020
#3
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Question: is there a way to do it without sterling numbers?

Guest Jan 18, 2020