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River Token Ways

Dynamic Programming Hard 16 views
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In River Token Ways you need the number of distinct combinations of token values that sum to a target total where each token value can be used unlimited times Order does not matter using 1 2 2 is the same combination regardless of arrangement This is a classic unbounded-combination counting problem not a permutation counter The most dependable solution is coin-change style dynamic programming Use a one-dimensional array where dp t stores ways to build total t Initialize dp 0 1 then process token values one by one updating totals from token value up to target This iteration order prevents overcounting by combination order and naturally handles unlimited reuse of each token type All additions must be modulo 1 000 000 007 for large result safety The judge includes empty token sets unreachable targets repeated token denominations and target zero behavior In this setup target zero always has exactly one valid way pick nothing Return a single integer count deterministic across runs This hard-level exercise strengthens your understanding of DP state design transition order and subtle counting correctness rules that frequently appear in real optimization and combinatorics workflows One common pitfall is iterating totals in the wrong direction which accidentally turns a combination problem into a permutation count Keep token iteration outermost and total iteration forward so each denomination contributes correctly without introducing order-sensitive duplicates in the final count Because this is a counting optimization-style challenge dynamic programming is usually

Examples

Example 1
Input
tokens = [1,2,5], target = 5
Output
4
Explanation

Four multisets reach the total: 5; 2+2+1; 2+1+1+1; and five copies of 1.

Example 2
Input
tokens = [2], target = 3
Output
0
Explanation

No combination of value 2 tokens can reach 3.

Example 3
Input
tokens = [10], target = 0
Output
1
Explanation

Choosing nothing is the single combination that meets the zero target.

Algorithm Flow

Recommendation Algorithm Flow for River Token Ways - Budibadu
Recommendation Algorithm Flow for River Token Ways - Budibadu

Best Answers

java
class Solution {
    public int count_token_combinations(int n, int[] tokens) {
        int[] dp = new int[n + 1];
        dp[0] = 1;
        for (int t : tokens) {
            for (int i = t; i <= n; i++) {
                dp[i] += dp[i - t];
            }
        }
        return dp[n];
    }
}