I jumped straight to recursion and it worked but was way too slow.
Start by clarifying the problem constraints and then present a dynamic programming solution. Define dp[i][s] as the number of ways to get sum s with i dice, and derive the recurrence dp[i][s] = sum_{f=1..k} dp[i-1][s-f]. Discuss time and space complexity, and mention optimizations like space reduction and combinatorial methods.
Pro tip: Always discuss edge cases and constraints upfront, and mention that the DP can be optimized to O(n*target) time and O(target) space, showing awareness of efficiency. Also, briefly note that for large n and k, a combinatorial inclusion-exclusion approach can be more efficient.
Ask about constraints: range of n, k, target, and whether dice are distinguishable. Confirm if the answer should be modulo a number (e.g., 10^9+7) and if target can be negative or zero.
Define dp[i][s] as the number of ways to get sum s using i dice. Recurrence: dp[i][s] = sum_{f=1..k} dp[i-1][s-f], with base case dp[0][0]=1.
Implement bottom-up DP, using a 1D array to save space. Iterate over dice and sums, using a sliding window or prefix sums to optimize the inner loop to O(1) per state.
Time complexity O(n*target) with optimization, space O(target). Handle edge cases: target < n or target > n*k returns 0; n=0 returns 1 if target=0 else 0.
Mention combinatorial inclusion-exclusion: number of solutions to sum = target with each die between 1 and k is sum_{j} (-1)^j C(n,j) C(target - k*j - 1, n-1). Also note generating functions.
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