Can anyone tell me how I prove this dp relation of selections of j object from i dp[i][j] = dp[i-1][j-1] + dp[i-1][j] if anyone can explain me the two states in the right hand side of the eqn it would be great. please explain the transitions and states in this problem.
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dp[i][j]=number of ways of choosing j elements out of i elements. First consider any ordering of the elements. Now consider the ith element in this ordering, dp[i][j] can be written as sum of two parts
That contains the ith element, so now you have taken 1 element and you need to take remaining j-1 elements from remaining i-1 elements ie dp[i-1][j-1].
That do not contain ith element so you need to choose all j elements from the remaining i-1 elements ie dp[i-1][j]