Step by Step solution about How to resolve the algorithm Count the coins step by step in the Mathematica / Wolfram Language programming language

The provided Wolfram code defines a function called CountCoins that counts the number of ways to make change for a given monetary amount using a list of coin denominations. Here's a detailed explanation of the code:

1. CountCoins[amount_, coinlist_] :=: This line defines the CountCoins function, which takes two arguments: amount, which is the monetary amount for which you want to count the number of ways to make change, and coinlist, which is a list of coin denominations available for making change.

2. (ways = ConstantArray[1, amount];: This line initializes an array ways with length amount and sets all its elements to 1. The ConstantArray function creates an array of the specified length, where each element has the given constant value. In this case, it creates an array of length amount, where each element is 1.

3. Do[For[j = coin, j <= amount, j++,: This line starts a Do loop over each coin denomination in the coinlist. The loop variable j represents the current coin denomination being considered.

4. If[ j - coin == 0,: Within the inner For loop, it checks if the current coin denomination (j) is equal to the amount to be changed (amount). If j is equal to amount, it means that a single coin of denomination j can exactly make up the desired amount, so it increments the corresponding element in the ways array by 1.

5. ways[[j]] ++,: If j is not equal to amount, it means that a single coin of denomination j cannot make up the desired amount exactly. In this case, it adds the value stored in ways[[j - coin]] to the current element in the ways array. This step accumulates the number of ways to make change using the previous coin denominations and the current coin denomination.

6. ]], ]: These closing brackets end the inner For loop and the outer Do loop.

7. ways[[amount]]: After the loops have completed, it returns the element in the ways array corresponding to the amount index. This value represents the total number of ways to make change for the given monetary amount using the provided coin denominations.

In summary, the CountCoins function uses dynamic programming to efficiently count the number of ways to make change for a given monetary amount using a specified list of coin denominations. It does this by populating an array ways to store the number of ways to make change for all possible amounts up to the given amount, and then returning the number of ways for the desired amount from the ways array.

CountCoins[amount_, coinlist_] := ( ways = ConstantArray[1, amount];
Do[For[j = coin, j <= amount, j++,
  If[ j - coin == 0,
    ways[[j]] ++,
    ways[[j]] += ways[[j - coin]]
]]
, {coin, coinlist}];
ways[[amount]])