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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.

 Jan 1, 2023

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 #1
avatar+1632 
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(a) Friends, a, b, c, d, and e.

a + b + c + d + e = 20. All friends are nonnegative, so by stars and bars we have (20 + 5 - 1 choose 5 - 1) = (24 choose 4) = 10626 ways to give out 20 identical stickers to his friends.

 Jan 1, 2023
 #2
avatar+1632 
+2

(b) By stars and bars, instead of (n + k - 1 choose k - 1), we have (n - 1 choose k - 1), so (19 choose 4) = 3876 ways.

 Jan 1, 2023

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