ellr's blog

Classical problem
By ellr, 20 months ago, translation, In English

Ive need this in some problems, but I can't figure out how to count it. I will be glad to useful ideas.

How many different arrays of length n are there of non-negative integers with sum of k; n <= 10^5, k <= 10^9

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20 months ago, # |
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https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

Stars and Bars

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20 months ago, # |
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We can solve this problem using combinatorics.

Consider the "stars and bars" method. Imagine you have k stars and n-1 bars. The stars represent the sum, and the bars divide these stars into n parts (each part corresponds to an element of the array).

The total number of ways to arrange the k stars and n-1 bars is the number of ways to choose (n-1) positions for the bars among (k+n-1) total positions.

This is a combinatorial problem and can be solved using the binomial coefficient, often denoted as "n choose k" or C(n, k), which is given by: C(n, k) = n! / (k!(n-k)!)

In this case, we want to calculate C(k+n-1, n-1). The result may be very large, so you might need to calculate it modulo some number (e.g., 10^9 + 7).

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    20 months ago, # ^ |
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    It's worth noting that given the size of $$$k$$$ in the given problem, using the factorial formula directly is likely to be too slow when computing the answer modulo, say, $$$10^9 + 7$$$.

    Instead, rewrite the formula without factorials, with a product of $$$(n - 1)$$$ terms in both the numerator and denominator — since $$$n$$$ is small, the two can be evaluated fast enough.

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