Select three (3) coins, but assume that there are 4 dimes, 4 nickels,
and 2 quarters.
In how many possible ways can the selection be made so that
the value of the coins is at least 25 cents?
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.
To find the number of possible ways to make a selection of three coins such that the value is at least 25 cents, we need to consider the different combinations of coins that can be chosen.
In this case, we have 4 dimes, 4 nickels, and 2 quarters. To ensure that the value is at least 25 cents, we need to consider combinations that include either a quarter or a combination of dimes and nickels.
Let's break it down into cases:
Case 1: Selecting a quarter:
Since we have 2 quarters, we can choose 1 quarter and then select 2 more coins from the remaining coins (dimes and nickels). The number of ways to do this is given by:
2 C 1 * (4 + 4) C 2 = 2 * 8 C 2
Case 2: Selecting only nickels and dimes:
We can select all three coins from either nickels or dimes, or we can select two coins from one type and one coin from the other type. The number of ways to do this is given by:
(4 C 3) + (4 C 2) * (4 C 1) + (4 C 1) * (4 C 2) = 4 + (6 * 4) + (4 * 6) = 4 + 24 + 24 = 52
Therefore, the total number of possible ways to make a selection of three coins such that the value is at least 25 cents is:
2 * 8 C 2 + 52 = 2 * (8! / (2! * (8 - 2)!)) + 52 = (2 * 28) + 52 = 56 + 52 = 108.