Given three positive integers N, L and R. The task is to find the number of ways to form an array of size N where each element lies in the range [L, R] such that the total sum of all the elements of the array is divisible by 2.
Input: N = 2, L = 1, R = 3
Possible arrays having sum of all elements divisible by 2 are
[1, 1], [2, 2], [1, 3], [3, 1] and [3, 3]
Input: N = 3, L = 2, R = 2
Approach: The idea is to find the count of numbers having remainder 0 and 1 modulo 2 separately lying between L and R. This count can be calculated as follows:
We need to count numbers between range having remainder 1 modulo 2
F = First number in range of required type
L = Last number in range of required type
Count = (L – F) / 2
cnt0, and cnt1 represents Count of numbers between range of each type.
Then, using dynamic programming we can solve this problem. Let dp[i][j] denotes the number of ways where the sum of first i numbers modulo 2 is equal to j. Suppose we need to calculate dp[i], then it will have the following recurrence relation: dp[i] = (cnt0 * dp[i – 1] + cnt1 * dp[i – 1]). First term represents the number of ways upto (i – 1) having sum remainder as 0, so we can place cnt0 numbers in ith position such that sum remainder still remains 0. Second term represents the number of ways upto (i – 1) having sum remainder as 1, so we can place cnt1 numbers in ith position to such that sum remainder becomes 0. Similarly, we can calculate for dp[i].
Final answer will be denoted by dp[N].
Below is the implementation of the above approach:
- Number of ways to form a heap with n distinct integers
- Count integers in the range [A, B] that are not divisible by C and D
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- Number of integers in a range [L, R] which are divisible by exactly K of it's digits
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- Count numbers in a range that are divisible by all array elements
- Find a non empty subset in an array of N integers such that sum of elements of subset is divisible by N
- Total ways of choosing X men and Y women from a total of M men and W women
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- Number of ways to sum up a total of N from limited denominations
- Total ways of selecting a group of X men from N men with or without including a particular man
- Count of sub-sets of size n with total element sum divisible by 3
- Different ways to represent N as sum of K non-zero integers
- Ways to write N as sum of two or more positive integers | Set-2
- Ways to write n as sum of two or more positive integers
- Count total divisors of A or B in a given range
- Sum of last digit of all integers from 1 to N divisible by M
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