#114 Medium Time O(n·m) Space O(n) Animated

Coin Change II

Count number of ways to make change.

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1 Predict
2 Trace
3 Explain
4 Implement

Pattern Two-dimensional dynamic programming

A table exposes how two changing prefixes or positions depend on smaller cases.

Approach: Dynamic Programming (2D)

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Core Insight

A table exposes how two changing prefixes or positions depend on smaller cases.

Common Pitfall

Initialize the first row and column before using neighboring cells.

Description

Count number of ways to make change.

Constraints

Time Complexity: O(n·m)
Space Complexity: O(n)

Solution — Dynamic Programming (2D)

TimeO(n·m) SpaceO(n)

Implementation Plan

Translate the invariant into these small, testable moves.

Define the row and column meaning.
Initialize base cases.
Fill cells in dependency order.
Read the answer from the target cell.

def change(amount, coins):
    dp = [0] * (amount + 1)
    dp[0] = 1
    
    for i in range(len(coins) - 1, -1, -1):
        nextDP = [0] * (amount + 1)
        nextDP[0] = 1
        for a in range(1, amount + 1):
            nextDP[a] = dp[a]
            if a - coins[i] >= 0:
                nextDP[a] += nextDP[a - coins[i]]
        dp = nextDP
        
    return dp[amount]

Best Time to Buy and Sell Stock with Cooldown

Target Sum

Coin Change II

Steps

Description

Constraints

Tags

Solution — Dynamic Programming (2D)