# Coin Combinations: Solving Part a) and Questions for Part b)

• Workout
In summary, to make 60 out of 20 1s, 2s, and 5s, you need to work with a total of 16 different combinations.
Workout

## Homework Statement

6. a) In your pocket you have some nickels, dimes, and quarters. There are 20 coins altogether and exactly twice as many dimes as nickels. The total value of the coins is $3.00. Find the number of coins of each type. b) Find all possible combinations of 20 coins (nickels, dimes, and quarters) that will make exactly$3.00.

I solved part a) and got 4 nickels, 8 dimes, 8 quarters.

I really don't know how to do (b). I could make a chart, but is there a mathematical procedure to doing this? I don't know how to put it in matrice form.

Workout said:

## Homework Statement

6. a) In your pocket you have some nickels, dimes, and quarters. There are 20 coins altogether and exactly twice as many dimes as nickels. The total value of the coins is $3.00. Find the number of coins of each type. b) Find all possible combinations of 20 coins (nickels, dimes, and quarters) that will make exactly$3.00.

I solved part a) and got 4 nickels, 8 dimes, 8 quarters.

I really don't know how to do (b). I could make a chart, but is there a mathematical procedure to doing this?
I don't believe there is, but it helps to think in a systematic way. For example, you can't have only nickels or only dimes, so that eliminates lots of possiblities. Also, you can't have only quarters, because that would leave you with too much money.

Start with the largest number of quarters for which you have a chance of getting \$3, and then see if some combination of nickels and dimes gives you the right amount.

Work you way down with 1 less quarter each time, looking at the number of nickels and dimes that will work.
Workout said:
I don't know how to put it in matrice form.
The word is matrix. Its plural is matrices. There is no word "matrice" in English. You can thank the Romans for this contribution to the complexity of English.

BTW, this is not a Calculus problem, so I'm moving it to the Precalc section.

b) reduces to making a total of 60 from twenty 1s, 2s and 5s.

Let the number of 5s be n. If that's odd then there must also be an odd number of 1s, so it splits into two cases. E.g. for n=2k, there's 60-10k to be made from 20-2k 2s and 1s. If there are m 2s we have 2m+(20-2k-m)=60-10k. After simplifying, you can quickly see what combinations are possible.

## 1. What is the purpose of solving coin combinations in part a)?

The purpose of solving coin combinations in part a) is to find all possible combinations of coins that add up to a given value. This can be useful for tasks such as making change or calculating the total value of a collection of coins.

## 2. How do you solve for coin combinations in part a)?

In part a), you can solve for coin combinations by using a recursive algorithm. This involves breaking down the problem into smaller subproblems and building up a solution from the results of these subproblems. For example, to find all combinations of coins that add up to 10 cents, you would first consider all combinations of coins that add up to 9 cents, and then add a 1 cent coin to each of these combinations.

## 3. What is the significance of the "order matters" condition in part b)?

Part b) of the coin combinations problem introduces the condition that the order in which the coins are selected matters. This means that, for example, selecting a 10 cent coin followed by a 5 cent coin is considered a different combination than selecting a 5 cent coin followed by a 10 cent coin. This condition is important because it changes the number of possible combinations and requires a different approach to solving the problem.

## 4. How can you modify the algorithm for part a) to handle the "order matters" condition in part b)?

To handle the "order matters" condition in part b), you can modify the algorithm used in part a) by keeping track of the coins that have already been selected in each combination. This ensures that the same coin is not selected more than once in a single combination, and therefore, the order of selection is maintained. Additionally, you may need to adjust the base case and recursive calls to account for this condition.

## 5. Can this problem be extended to include different types of coins or a larger coin set?

Yes, this problem can be extended to include different types of coins or a larger coin set. The approach for solving the problem would remain the same, but the base cases and recursive calls may need to be modified to handle the new types of coins or larger coin values. With a larger coin set, the number of possible combinations will also increase, making the problem more complex to solve.

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