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Chapter 34: Problem 1093

A boy has in his pocket a penny, a nickel, a dime, and a quarter. How many different sums of money can he take out if he removes one or more coins?

Short Answer

Expert verified
The boy can take out a total of 15 different sums of money from his pocket if he removes one or more coins.

Step by step solution

01

Identifying coin values

First, let's assign the values to each of the four coins the boy has in his pocket: 1. Penny: \(1垄\) 2. Nickel: \(5垄\) 3. Dime: \(10垄\) 4. Quarter: \(25垄\)
02

Counting single coin combinations

If the boy takes out just one coin, there are 4 unique sums of money he can take out: \(1垄\), \(5垄\), \(10垄\), and \(25垄\).
03

Counting two-coin combinations

If the boy takes out two coins, let's count the unique sums of money as follows: 1. Penny + Nickel = \(1垄 + 5垄 = 6垄\) 2. Penny + Dime = \(1垄 + 10垄 = 11垄\) 3. Penny + Quarter = \(1垄 + 25垄 = 26垄\) 4. Nickel + Dime = \(5垄 + 10垄 = 15垄\) 5. Nickel + Quarter = \(5垄 + 25垄 = 30垄\) 6. Dime + Quarter = \(10垄 + 25垄 = 35垄\) So, there are 6 unique sums of money when the boy takes out two coins.
04

Counting three-coin combinations

If the boy takes out three coins, let's count the unique sums of money as follows: 1. Penny + Nickel + Dime = \(1垄 + 5垄 + 10垄 = 16垄\) 2. Penny + Nickel + Quarter = \(1垄 + 5垄 + 25垄 = 31垄\) 3. Penny + Dime + Quarter = \(1垄 + 10垄 + 25垄 = 36垄\) 4. Nickel + Dime + Quarter = \(5垄 + 10垄 + 25垄 = 40垄\) There are 4 unique sums of money when the boy takes out three coins.
05

Counting a four-coin combination

If the boy takes out all four coins, there is only one unique sum of money: Penny + Nickel + Dime + Quarter = \(1垄 + 5垄 + 10垄 + 25垄 = 41垄\)
06

Calculating the total number of combinations

Now, let's sum up all the unique sums of money from the combinations of one or more coins: \(4 \text{(single coin)} + 6 \text{(two-coin)} + 4 \text{(three-coin)} + 1 \text{(four-coin)} = 15\) In conclusion, the boy can take out a total of 15 different sums of money from his pocket if he removes one or more coins.

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