We are given N items which are of total K different colors. Items of the same color are indistinguishable and colors can be numbered from 1 to K and count of items of each color is also given as k1, k2 and so on. Now we need to arrange these items one by one under a constraint that the last item of color i comes before the last item of color (i + 1) for all possible colors. Our goal is to find out how many ways this can be achieved.
Input : N = 3 k1 = 1 k2 = 2 Output : 2 Explanation : Possible ways to arrange are, k1, k2, k2 k2, k1, k2 Input : N = 4 k1 = 2 k2 = 2 Output : 3 Explanation : Possible ways to arrange are, k1, k2, k1, k2 k1, k1, k2, k2 k2, k1, k1, k2
We can solve this problem using dynamic programming. Let dp[i] stores the number of ways to arrange first i colored items. For one colored item answer will be one because there is only one way. Now Let’s assume all items are in a sequence. Now, to go from dp[i] to dp[i + 1], we need to put at least one item of color (i + 1) at the very end, but the other items of color (i + 1) can go anywhere in the sequence. The number of ways to arrange the item of color (i + 1) is combination of (k1 + k2 .. + ki + k(i + 1) – 1) over (k(i + 1) – 1) which can be represented as (k1 + k2 .. + ki + k(i + 1) – 1)C(k(i + 1) – 1). In this expression we subtracted one because we need to put one item at the very end.
In below code, first we have calculated the combination values, you can read more about that from here. After that we looped over all different color and calculated the final value using above relation.
This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
- Count the number of ways to tile the floor of size n x m using 1 x m size tiles
- Minimum number of jumps to reach end
- Total number of possible Binary Search Trees and Binary Trees with n keys
- Efficient program to print all prime factors of a given number
- Program for nth Catalan Number
- Count number of binary strings without consecutive 1's
- Count ways to reach the n'th stair
- Count number of ways to reach a given score in a game
- How to print maximum number of A's using given four keys
- Count possible ways to construct buildings
- Find minimum number of coins that make a given value
- Minimum number of squares whose sum equals to given number n
- Find number of solutions of a linear equation of n variables
- Total number of non-decreasing numbers with n digits
- Count total number of N digit numbers such that the difference between sum of even and odd digits is 1