Exploring Coin Selections for a Quarter Value Requirement

Let's delve into a combinatorial problem involving coin selection. Specifically, we are tasked with selecting three coins from a collection consisting of 4 dimes, 4 nickels, and 2 quarters, with the condition that the total value of the selected coins must be at least 25 cents. This problem can be effectively analyzed through a detailed breakdown of possible selections and their respective values.

Understanding Coin Values

First, let's establish the values of the coins available:

Dime (D): 10 cents Nickel (N): 5 cents Quarter (Q): 25 cents

Total Value Requirement

The challenge is to ensure that the combination of selected coins adds up to at least 25 cents. To achieve this, we can explore different combinations based on the inclusion of quarters, as a quarter alone meets the requirement of 25 cents.

Case Analyses

Case 1: Selecting 0 Quarters

In this scenario, we are solely selecting dimes and nickels. We need to find combinations where the total value is at least 25 cents:

3 Dimes (30 cents): DDD 2 Dimes and 1 Nickel (25 cents): DD N 1 Dime and 2 Nickels (20 cents): D NN (Does not meet the requirement) 3 Nickels (15 cents): NNN (Does not meet the requirement)

The valid combinations in this case are:

DDD DD N

This results in a total of 2 ways.

Case 2: Selecting 1 Quarter

With one quarter, we need to add two more coins from the remaining pool of dimes and nickels. Since the quarter already contributes 25 cents, we only need to ensure that the remaining two coins do not bring the total below 25 cents:

1 Quarter and 2 Dimes: QDD 1 Quarter, 1 Dime, and 1 Nickel: QDN 1 Quarter and 2 Nickels: QNN

These valid combinations yield a total of 3 ways.

Case 3: Selecting 2 Quarters

In this case, we select two quarters and one additional coin. The additional coin can be either a dime or a nickel, as the two quarters alone already meet the 25-cent requirement:

2 Quarters and a Dime: QQD 2 Quarters and a Nickel: QQN

This results in another 2 valid ways.

Total Combinations

Summing up the valid combinations from all cases, we get:

Case 1: 2 ways Case 2: 3 ways Case 3: 2 ways

Total number of valid ways: 2 3 2 7 ways.

Conclusion

The total number of ways to select three coins such that their value is at least 25 cents is 7 ways.

Advanced Analysis for Permutations and Probability

For a more detailed analysis, considering permutations and probability, we can approach it as follows:

Total permutations for selecting 3 coins out of 10: 10P3 10! / (10 - 3)! 720 Total valid permutations: 7 (as determined above)

This would result in a probability of:

2/720 1/360 ≈ 0.278%

However, as per the problem's statement and the context of combinatorial selection, the focus is primarily on the number of valid selections (7 ways) without considering probability.