To determine the number of ways to select three coins with a value of at least 25 cents, we can use a combination of cases:
Case 1: Selecting only quarters.
Since there are 2 quarters available, we can only select them both, as we need at least 25 cents. Therefore, there is only 1 way to select two quarters.
Case 2: Selecting one quarter and two other coins.
We need to select one quarter, which leaves us with two coins to choose from (4 dimes and 4 nickels). The possible combinations are:
- Selecting 1 quarter and 2 dimes: 2 ways (2 dimes left)
- Selecting 1 quarter and 1 dime + 1 nickel: 4 ways (2 dimes and 4 nickels left)
- Selecting 1 quarter and 2 nickels: 1 way (4 nickels left)
Case 3: Selecting three coins that do not include quarters.
We need to select three coins from 4 dimes and 4 nickels. We can calculate this using combinations. The possible combinations are:
- Selecting 3 dimes: C(4, 3) = 4 ways (1 dime left)
- Selecting 2 dimes + 1 nickel: C(4, 2) * C(4, 1) = 6 ways (2 dimes and 3 nickels left)
- Selecting 1 dime + 2 nickels: C(4, 1) * C(4, 2) = 12 ways (3 dimes and 2 nickels left)
- Selecting 3 nickels: C(4, 3) = 4 ways (4 dimes left)
Now, let's calculate the total number of ways in all cases:
Total ways = Case 1 + Case 2 + Case 3
= 1 + (2 + 4 + 1) + (4 + 6 + 12 + 4)
= 1 + 7 + 26
= 34
Therefore, there are 34 possible ways to select three coins with a value of at least 25 cents.