QZ wrote:

Flor is choosing three of five colors of paint to use for her art project at school. Two of the colors, Green and Yellow, cannot both be selected. How many different ways can Flor choose the colors for her project?

A. 7

B. 9

C. 10

D. 13

E. 17

There are three cases: 1) green is one of the five colors chosen, but yellow isn’t, 2) yellow is one of the five colors chosen, but green isn’t, and 3) neither green nor yellow is chosen. Let’s analyze each case.

Case 1: Green is one of the five colors chosen, but yellow isn’t.

If green is chosen but yellow isn’t, then we have to choose 2 more colors from the 3 remaining colors. The number of ways to do that is 3C2 = 3.

Case 2: Yellow is one of the five colors chosen, but green isn’t.

This is analogous to case 1, so there are 3 ways for this case..

Case 3: Neither green nor yellow is chosen.

If neither color is chosen, then we have to choose 3 colors from the 3 remaining colors. The number of ways to do that is 3C3 = 1.

Thus, the total number of ways Flor can choose the colors for her project is 3 + 3 + 1 = 7.

Alternate Solution:

We can use the formula:

Total number of ways to pick 3 colors = number of ways where yellow and green are both included + number of ways where yellow and green are not both included

Since we are choosing 3 colors from 5 available colors, there are 5C3 = (5 x 4)/2 = 10 ways of doing this when there are no restrictions.

The number of ways where yellow and green are both included can be found easily by observing that yellow and green occupy two of the three slots; any one of the remaining three colors can occupy the final slot. So, there are 3 ways to choose colors where yellow and green are both included.

Thus, the number of ways to pick colors where yellow and green are not included together is 10 - 3 = 7.

Answer: A

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