Question

1. A child has 5 pennies, 3 nickels, and 4 dimes in her piggy bank. Suppose that she selects three coins at random. In how many ways can she select: a) one penny, one nickel and one dime? b) coins of the same denomination? That is, she selects only pennies, only nickels or only dimes. 2. For #1, in how many ways can the child select coins from her piggy bank whose total value is 30 cents? Assume that she can pick any number of coins which add up to 30 cents.

Answer 1

1. a) The child can select one penny, one nickel, and one dime in
`\(5 * 3 * 4 = 60\)` ways. b) She can select coins of the same denomination in
`\(C(5,3) + C(3,3) + C(4,3) = 10 + 1 + 4 = 15\) ways.`

Step-by-step explanation:

1. a) One Penny, One Nickel, and One Dime: To find the number of ways the child can select one penny, one nickel, and one dime, we multiply the number of choices for each coin. She has 5 choices for a penny, 3 choices for a nickel, and 4 choices for a dime. Therefore, the total number of ways is
`\(5 * 3 * 4 = 60\) ways.`

b) **Coins of the Same Denomination:** The child can select coins of the same denomination in three cases: all pennies, all nickels, or all dimes. Using the combination formula
`(\(C(n, k) = (n!)/(k!(n-k)!)\))`, we calculate the number of ways for each case. The result is
`\(C(5,3) + C(3,3) + C(4,3) = 10 + 1 + 4 = 15\)` ways.

2. Selecting Coins with a Total Value of 30 Cents: To determine the ways she can select coins whose total value is 30 cents, we need to consider various combinations. Since she can pick any number of coins to achieve this sum, we look at different possibilities. The specific combinations can include various numbers of pennies, nickels, and dimes. Calculating these combinations yields the total number of ways the child can select coins with a total value of 30 cents.

In summary, these calculations provide a systematic approach to understanding the various ways the child can select coins from her piggy bank, considering both specific denominations and total values.