Partitioned Sums
How many ways can we express the number 23 as the sum of 13 non-negative integers, if repetition is allowed?
Details & Assumptions
All orderings of a sums count. For example 1+2 and 2+1 are considered different ways of representing the number 3 as the sum of 2 non-negative integers.
The answer is 834451800.
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1 solution
Let there be 13 bins such that you place 23 (+1) counters. Since repetition is allowed, you may place multiple counters in a bin. This illustration translates the problem into finding combinations with repetitions allowed (a.k.a stars and bars). So we evaluate ( 2 3 + 1 3 − 1 2 3 ) = 8 3 4 4 5 1 8 0 0 .