there are 12 in the opera half of them must where long baggy pant. ¼ long straight pants and the rest are short pants. in how many ways can the pants allocated them ( answer serious ) ( show solution )
First, we need to determine how many people must wear long baggy pants. We are given that half of the 12 people must wear long baggy pants, which is 6 people.
Next, we need to determine how many ways we can allocate the pants for the 6 people who must wear long baggy pants. Since all 6 of them must wear long baggy pants, we don't need to make any choices for them.
For the remaining 6 people, we need to allocate their pants among 3 options: 1/4 of them must wear long straight pants, and the rest are short pants. We can use the multiplication principle to find the total number of ways we can allocate their pants.
Let's start with the people who will wear long straight pants. We need to choose 1/4 of the 6 people, which is 1. We can choose this person in 6 ways.
For the people who will wear short pants, we need to choose 5 of the 6 people. We can choose these people in C(6,5) = 6 ways.
Using the multiplication principle, the total number of ways we can allocate the pants is:
6 x 6 x C(6,5) = 6 x 6 x 6 = 216 ways.
Therefore, there are 216 ways to allocate the pants for the 12 people in the opera.
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Step-by-step explanation:
First, we need to determine how many people must wear long baggy pants. We are given that half of the 12 people must wear long baggy pants, which is 6 people.
Next, we need to determine how many ways we can allocate the pants for the 6 people who must wear long baggy pants. Since all 6 of them must wear long baggy pants, we don't need to make any choices for them.
For the remaining 6 people, we need to allocate their pants among 3 options: 1/4 of them must wear long straight pants, and the rest are short pants. We can use the multiplication principle to find the total number of ways we can allocate their pants.
Let's start with the people who will wear long straight pants. We need to choose 1/4 of the 6 people, which is 1. We can choose this person in 6 ways.
For the people who will wear short pants, we need to choose 5 of the 6 people. We can choose these people in C(6,5) = 6 ways.
Using the multiplication principle, the total number of ways we can allocate the pants is:
6 x 6 x C(6,5) = 6 x 6 x 6 = 216 ways.
Therefore, there are 216 ways to allocate the pants for the 12 people in the opera.