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counting and probability intermediate
Problem
Liz has 11 marbles of different colors. In how many ways can she choose 3 of the marbles, if she refuses to choose the green one and the purple one at the same time?
Solution
We can divide this into cases:
Case 1: Liz chooses the green marble, and not the purple marble. In this case, Liz must choose 2 more marbles from the remaining 9 marbles (since she won't choose the purple marble). So, there are choices in this case.
Case 2: Liz chooses the purple marble, and not the green marble. As in Case 1, Liz must choose 2 more marbles from the remaining 9 marbles. So, there are choices in this case.
Case 3: Liz chooses neither the green marble nor the purple marble. In this case, Liz must choose three marbles from the nine remaining marbles, and there are choices in this case.
The total number of possible choices is .
Alternatively, there are ways to select three marbles. Of those, ways contain both the purple marble and the green marble. Therefore, there are ways to select three marbles such that purple and green marbles are not both chosen.
Case 1: Liz chooses the green marble, and not the purple marble. In this case, Liz must choose 2 more marbles from the remaining 9 marbles (since she won't choose the purple marble). So, there are choices in this case.
Case 2: Liz chooses the purple marble, and not the green marble. As in Case 1, Liz must choose 2 more marbles from the remaining 9 marbles. So, there are choices in this case.
Case 3: Liz chooses neither the green marble nor the purple marble. In this case, Liz must choose three marbles from the nine remaining marbles, and there are choices in this case.
The total number of possible choices is .
Alternatively, there are ways to select three marbles. Of those, ways contain both the purple marble and the green marble. Therefore, there are ways to select three marbles such that purple and green marbles are not both chosen.
Final answer
156