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How many ways are there to distribute 12 identical candies among 5 children with each child having at most 5 candies?

 Jan 1, 2023
 #1
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The number of ways is 7644.

 Jan 1, 2023
 #2
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Let the children be a, b, c, d, and e since they are different.

a + b + c + d + e = 12. By stars and bars if each variable is nonnegative, we have (12 + 5 - 1 choose 5 - 1) = 16 choose 4 = 1820 ways.

However we overcounted the ways for when the variables are more than 5. But these cases are easy to count:

Case 1: There was a 6.

a + b + c + d = 6. Stars and bars = 9 choose 3 = 84 ways, multiply this by 5 to get the cases when d, c, b, and a are 6 = 420 ways.

Case 2: There was a 7. 

... 

Then once you finish up the cases, all you have to do is take 1820, and subtract the overcounts. (The answer is not >1400)

 Jan 1, 2023
 #3
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1 - [12 + 5 - 1] C [5 - 1] ==1820 ways without restrictions

 

2 - 1820  -  [listfor(n, 1, 7, ((10 - n) nCr (4 - 1))* 5)]

 

3 - 1820  -  [420+ 280+ 175+ 100+ 50+ 20+ 5] + 10 [Because 6 occurs twice 10 times]

 

4 - 780 ways to distribute 12 candies to 5 kids each having at most 5 candies.

 Jan 4, 2023

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