Pi Day Pie Distribution
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Count the nondecreasing distributions of n pie pieces among k people with each person getting at least one piece.
- Level
Medium5 of 10
- Topics
- Dynamic programming, Combinatorics
- Solved
- No attempts yet
Problem
You may know that March 14 is known as “π-day”, since 3.14 (which is the third month and fourteenth day) is a good approximation of π.
Mathematicians celebrate this day by eating pie.
Suppose that you have n pieces of pie, and k people who are lined up for pieces of pie. All n pieces of pie will be given out. Each person will get at least one piece of pie, but mathematicians are a bit greedy at times. So, they always get at least as many of pieces of pie as the person in front of them.
For example, if you have 8 pieces of pie and 4 people in line, you could give out pieces of pie in the following five ways (with the first person in line being the first number in the list): [1, 1, 1, 5], [1, 1, 2, 4], [1, 1, 3, 3], [1, 2, 2, 3], [2, 2, 2, 2].
Notice that if k = n, there is only one way to give out the pieces of pie: every person gets exactly one piece. Also, if k = 1, there is only one way to give out the pieces of pie: that single person gets all the pieces.
Write a program that determines the number of ways that the pieces of pie can be given out.
Input
The first line of input is the integer number of pieces of pie, n (1 ≤ n ≤ 250). The second line of input is the integer k which is the number of people in line (1 ≤ k ≤ n).
For at least 20% of the marks for this problem, n ≤ 9. For at least 50% of the marks for this problem, n ≤ 70. For at least 85% of the marks for this problem, n ≤ 120.
Output
The output will consist of a single integer which is the number of ways that the pieces of pie can be distributed. The output is guaranteed to be less that 231.