From 10 people, how many ways can we form a committee of seven people consisting of a president, two (equivalent) vice presidents, and four (equivalent) regular members?

Question

From 10 people, how many ways can we form a committee of seven people consisting of a president, two (equivalent) vice presidents, and four (equivalent) regular members?

Answer 1

An alternative to Jeff's solution.

From 10 people, we are selecting 1 person for group a, 2 people for group b, 4 people for group c, and 3 for group d (the non-selected group). Then, the number of ways that the people can be selected is

10!/(1!2!4!3!) Think of the coefficient for a multinomial distribution.

Answer 2

Hi there!

To review the problem, we have a group of 10 people. We need to pick 7 of these people, specifically:

1. One president
2. Two vice presidents
3. Four members

We will tackle these criteria in order. First, we will choose the one president. So, how many ways can you choose one president from a group of ten people? That's simple, you can do this in ten ways. There are ten people, you can pick any one of them to be president giving ten possible choices.

Now that we have chosen a president, we will calculate how many ways we can choose a vice president. Since we have chosen some president, we now only have a pool of nine people to choose from.

When choosing some items from a group, it is important to note whether the order we choose these items matters or not. In this case, order doesn't matter as the vice presidents are equivalent, so we use a combination rather than a permutation. So, there are 9 choose 2, or 9! / (2! * 7!) which I will leave to you to calculate.

In a similar vein, you now need to choose four regular members from the seven members which remain after selecting the president and vice presidents. I'll leave it to you to calculate this - it follows the same procedure as choosing the vice presidents.

Finally, once you have each of these results, you can apply the product rule to find out the total number of ways to form this committee from the ten students. In other words, multiplying each of the above numbers together will give you your final answer. Good luck!