Compute Total Number of Distinct Ways to Reach Target

The DP problems belonging to this category, in its simplest form, looks like below or some variation of it:
  • Given a target find a number of distinct ways to reach the target.

The solution for this kind of problems, in a generalized form, would often look like below:
  • Sum all possible ways to reach the current state.

    distinctWays[i] = distinctWays[i-1] + distinctWays[i-2] + ... + distinctWays[i-k], where (i - 1), (i - 2), ... , (i - K) are all the directly immediate states from where i can be reached.
  • Overall the solution would look like:
    for (int curr = 1; curr <= target; curr++) {
       for (int k = 0; k < waysToReachCurrentTarget.size(); k++) {
        dp[i] += dp[waysToReachCurrentTarget[k]];
    return dp[target];

The below problem would give you a basic idea about this kind of problems:

Problem Statement:

You are climbing a stair case. It takes n steps to reach to the top.
Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top?
Example 1:
Input: 2
Output: 2
Explanation: There are two ways to climb to the top.
1. 1 step + 1 step
2. 2 steps
Example 2:
Input: 3
Output: 3
Explanation: There are three ways to climb to the top.
1. 1 step + 1 step + 1 step
2. 1 step + 2 steps
3. 2 steps + 1 step

Solution in Java and Python:

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