Three coins are selected from 10 coins: 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?

I know that the total outcomes equals 120, but how do I find how many of these have the value of at least 25 cents?

The answer is 92. Anyone have idea how to solve this or why the answer is 92? Thanks in advance!

Consider the cases which do not add up to 25, they would be

DNN and NNN
DNN -- C(4,1)C(4,2) = 24
NNN -- C(4,3 = 4
so 28 cases are not allowed
leaving 120 - 28 = 92

another way:

You have to list the possible outcomes , then evaluate each one, finally add up the allowable cases

QQQ - C(4,1) = 4
QQD - C(2,2)C(4,1) = 4
QNN - C(2,1)C(4,2) = 12
QDD - .... = 12
QND - .... = 32
DDD - .... = 4
DDN - .... = 24
DNN gives us < 25 cents
NNN gives us < 25 cents.

the total of the above is 92

To find the number of ways the selection can be made so that the value of the coins is at least 25 cents, you can use the concept of combinations.

Since you want to select three coins, you can start by looking at the various possibilities based on the combination of coins that would give you a value of at least 25 cents.

First, consider the cases where you have exactly one quarter (25 cents). You have 2 quarters to choose from. Once you have selected a quarter, you need to select two more coins from the remaining 8 coins (4 dimes and 4 nickels). This can be done in C(8, 2) = 28 ways.

Next, consider the cases where you have exactly two quarters. You have 2 quarters to choose from, and once you have selected two quarters, you need to select one more coin from the remaining 8 coins. This can be done in C(8, 1) = 8 ways.

Lastly, consider the case where you have three quarters. You have 2 quarters to choose from, but once you have selected three quarters, there are no more coins left to select from.

Therefore, the total number of ways to select three coins with a value of at least 25 cents is 28 + 8 + 1 = 37.

Hence, there are 37 possible ways to make the selection so that the value of the coins is at least 25 cents.