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(a) Find the number of ways that Magnus can give out 20 identical stickers to 5 of his friends.  (Not everyone has to get a sticker.)

 

(b) Find the number of ways that Magnus can give out 20 identical stickers to 5 of his friends, if every friend gets at least one sticker.

 Dec 29, 2022
 #1
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Partitions of 20, into at most 5 parts:

 

192  ways.

 

20 + 0 + 0 + 0 + 0

19 + 1 + 0 + 0 + 0

18 + 2 + 0 + 0 + 0

18 + 1 + 1 + 0 + 0

17 + 3 + 0 + 0 + 0

........and so on

 Dec 29, 2022
 #2
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I think the above answer is correct for IDENTICAL and IDENTICAL distribution. But 5 friend are not considered IDENTICAL but DISTINCT. The following is the more accurate formula:

 

[20 + 5 - 1] C 5 ==24 C 5 ==42,504 - number of ways.

 Dec 29, 2022
 #3
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[20  - 1] C [5 - 1] ==19 C 4 ==3,876 - number of ways.

 Dec 29, 2022

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