How Many Combinations of Digits 0-7 Sum to 7?

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The discussion focuses on finding the number of combinations of the digits 0 through 7 that sum to 7, using exactly three digits with repetition allowed. The procedure involves computing the expression (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)^3 and identifying the coefficient of x^7 in the resulting polynomial. This coefficient represents the total permutations of the digits that sum to 7, which can then be divided by 3! to obtain the number of unique combinations.

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MartinV05
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When you have combinations where digits are 0,1,2...,m, meaning we have n=m+1 and k, is there a way to see how much of them sum up to a given number? For the sake of simplicity I have the digits 0,1,2...,7 (so n=8), and k=3. I need to find how much of these combinations WITH repetition sum up to 7. Is there a formula for this?
 
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By sum up, I mean the sum of all 3 digits in each combination needs to be equal to 7.
 
MartinV05 said:
Is there a formula for this?

I don't know a formula, but there is a procedure - or at least a concise way to phrase the problem.

Compute

(1 + x + x^2 + x^3 +x^4 + x^5 + x^6 + x^7)^3 = ?

and then look at the coefficient of x^7 in the answer. The coefficient counts the number of permutations of the numbers 0,1,2...7 that add to 7. To get combinations, divide that by 3!.
 

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