van_persie9's blog

By van_persie9, history, 22 months ago, In English

Recently I came across a question in which we had to find the no. of ways to distribute N balls into 3 boxes such that the number of box getting maximum no. of balls is exactly 1.

For example

  • if N=2, ans is 2 : {2,0,0},{0,2,0},{0,0,2}

  • if N=3, ans is 9 : {3,0,0},{0,3,0},{0,0,3},{3,2,1},{3,1,2},{1,3,2},{2,3,1},{1,2,3},{2,1,3}

Now, since the number of boxes here is only 3, I was able to solve the question by observation and basic maths(Arith. Prog.) My query is how to solve the above question if the number of boxes is K?

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22 months ago, # |
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This can be calculated in $$$O((N + K) \log N)$$$. Let $$$\binom{a}{b} = 0$$$ when $$$a < b$$$. Using inclusion-exclusion, we get that the answer is

$$$K \sum_{s = 1}^{N} \sum_{c = 0}^{K-1} (-1)^{c} \binom{K-1}{c} \binom{K-1 + N-s - sc}{K-1}$$$

since $$$K$$$ is the number of ways to select the maximum element, we loop over $$$s$$$ which will be its value, and

$$$\sum_{c = 0}^{K-1} (-1)^{c} \binom{K-1}{c} \binom{K-1 + N-s - sc}{K-1}$$$

calculates the number of ways to select values for the other numbers such that no other number is $$$\geqslant s$$$. Changing the order of summation we get

$$$\sum_{c = 0}^{K-1} (-1)^{c} K \binom{K-1}{c} \sum_{s = 1}^{\left\lfloor\frac{N}{c+1}\right\rfloor} \binom{K-1 + N - s(c+1)}{K-1}$$$

Which can be directly calculated in $$$O((N + K) \log N)$$$ by precalculating factorials to compute the binomial terms in $$$O(1)$$$

22 months ago, # |
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Hey, isn’t it problem from ongoing contest? Why you didn’t asked after end of the contest?