Most of the problems we have looked at asked for either the maximum, minimum, or longest of something. However, it is also very common for a DP problem to ask for the number of distinct ways to do something. Climbing Stairs asked us to find the number of ways to climb to the top of the stairs.
Another term used to describe this class of problems is "counting DP".
What are the differences with counting DP? With the maximum/minimum problems, the recurrence relation typically involves a max() or min() function. This is true for all types of problems we have looked at - iteration, multi-dimensional, etc. With counting DP, the recurrence relation typically just sums the results of multiple states together. For example, in Climbing Stairs, the recurrence relation was dp(i) = dp(i - 1) + dp(i - 2). There is no max() or min(), just addition.
Another difference is in the base cases. In most of the problems we have looked at, if the state goes out of bounds, the base case equals 0. For example, in the Best Time to Buy and Sell Stock questions, when we ran out of transactions or ran out of days to trade, we returned 0 because we can't make any more profit. In Longest Common Subsequence, when we run out of characters for either string, we return 0 because the longest common subsequence of any string and an empty string is 0. With counting DP, the base cases are often not set to 0. This is because the recurrence relation usually only involves addition terms with other states, so if the base case was set to 0 then you would only ever add 0 to itself. Finding these base cases involves some logical thinking - for example, when we looked at Climbing Stairs - we reasoned that there is 1 way to climb to the first step and 2 ways to climb to the second step.