Compute Total Number of Distinct Ways to Reach Target

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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[i1] + distinctWays[i2] + ... + distinctWays[ik]
, 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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Abhishek Dey
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Microsoft  University of Florida
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