Finding the Distribution of Balls in Bins with Limited Capacity

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To find the distribution of 15 balls in 8 bins, each with a maximum capacity of 4, one must determine the number of combinations that satisfy these constraints. The problem requires finding non-negative integer solutions to the equation where each bin holds between 1 and 4 balls, totaling 15. A simplified formula for this distribution is sought, particularly one that counts the occurrences of each integer from 1 to 4 in the combinations. The discussion emphasizes the need for a systematic approach to calculate valid distributions efficiently. Understanding these constraints is crucial for solving the problem effectively.
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I have a 15 balls and 8 bins. Each bin can only hold 4 balls. If throw the 15 balls in the bins (random) and all the balls land in the bins, how do i find the distribution of balls/bin. eg. #4 bins with 3 balls, #1 bin with 1 ball etc.

Thanks!
 
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You will need to find the total number of ways that you can have 8 numbers, each at least 1, none larger than 4, that add to 15.
 
Yes. I should have mentioned that I figure that part out but was wondering if there was a simplified formula to do the same? Because, I would also have to be able to count the number of times each integer between 1 and 4 occurs in the this combination.

Any help would be appreciated.
 
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