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Chapter 22: Problem 517

How many different sums of money can be obtained by choosing two coins from a box containing a penny, a nickel, a dime, a quarter, and a half dollar?

Short Answer

Expert verified
There are 10 different sums of money that can be obtained by choosing two coins from a box containing a penny, a nickel, a dime, a quarter, and a half dollar.

Step by step solution

01

List the distinct pairs of coins

First, we need to list down all the distinct pairs of coins that can be chosen from the box. There are five coins in total: penny, nickel, dime, quarter, and half dollar. So we need to find all the distinct pairs of these five coins.
02

Calculate the number of unique pairs

The number of unique pairs that can be created from the five coins can be calculated using the combination formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!}, \] where n is the total number of elements, r is the size of each subset, n! is the factorial of n, and r! is the factorial of r. In this case, n = 5 (the five different coins) and r = 2 (since we are choosing two coins at a time). So, the number of unique pairs can be calculated as: \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{120}{2(6)} = \frac{120}{12} = 10. \]
03

List down the unique pairs of coins and their sums

Now, we can list out the 10 distinct possible pairs and their respective sums: 1. Penny & Nickel: 1 cent + 5 cents = 6 cents 2. Penny & Dime: 1 cent + 10 cents = 11 cents 3. Penny & Quarter: 1 cent + 25 cents = 26 cents 4. Penny & Half Dollar: 1 cent + 50 cents = 51 cents 5. Nickel & Dime: 5 cents + 10 cents = 15 cents 6. Nickel & Quarter: 5 cents + 25 cents = 30 cents 7. Nickel & Half Dollar: 5 cents + 50 cents = 55 cents 8. Dime & Quarter: 10 cents + 25 cents = 35 cents 9. Dime & Half Dollar: 10 cents + 50 cents = 60 cents 10. Quarter & Half Dollar: 25 cents + 50 cents = 75 cents
04

Count the number of distinct sums

By analyzing the list created in Step 3, we can observe that all 10 sums are unique, meaning that there are no repeated sums. Thus, there are 10 different sums of money that can be obtained by choosing two coins from the box.

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