Coin Change Total Number of Ways in DSA C - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main goal of the Coin Change Total Number of Ways problem?

To find how many different ways we can make a target amount using given coin denominations, where order does not matter.

beginner

In the Coin Change problem, why do we use dynamic programming?

Because it helps us build solutions for smaller amounts step-by-step and reuse those results to find the total ways for the target amount efficiently.

beginner

What does the dp array represent in the Coin Change Total Number of Ways solution?

dp[i] stores the number of ways to make the amount i using the given coins.

intermediate

How do we update the dp array when processing each coin?

For each coin, we add dp[j - coin] to dp[j] for all j from coin to target amount, meaning we add ways that include the current coin.

intermediate

Why does the order of loops matter in the Coin Change Total Number of Ways problem?

Using coins in the outer loop and amounts in the inner loop ensures combinations are counted without duplicates, as it fixes coin order and avoids permutations.

What does dp[0] represent in the Coin Change problem?

A Total amount of coins

B Number of coins used

C Number of ways to make amount 0, which is 1

D Zero ways to make amount 0

There is exactly one way to make amount 0: by choosing no coins.

If coins = [1, 2] and amount = 3, how many ways are there to make 3?

A 2

B 3

C 1

D 4

Ways: (1+1+1), (1+2), (2+1) but order does not matter, so (1+2) and (2+1) count as one. Total 2 ways.

Which loop order correctly counts combinations without duplicates?

A Loop order does not matter

B Amounts outer loop, coins inner loop

C Both orders count duplicates

D Coins outer loop, amounts inner loop

Coins outer loop ensures combinations are counted once by fixing coin order.

What is the time complexity of the Coin Change Total Number of Ways dynamic programming solution?

A O(n * amount)

B O(amount^2)

C O(n^2)

D O(n + amount)

Where n is number of coins and amount is the target sum.

What initial value should dp array have before processing coins?

A All negative ones

B All zeros except dp[0] = 1

C All zeros

D All ones

dp[0] = 1 because there is one way to make amount 0; others start at 0.

Explain how the dynamic programming array is built to solve the Coin Change Total Number of Ways problem.

Think about how smaller amounts help build solutions for bigger amounts.

Describe why the order of loops matters when counting the total number of ways to make change.

Consider how fixing coin order prevents counting the same combination multiple times.