Q24. - Algebra 2

Question

The members of the Math Club need to elect a president and a vice-president. They determine that there are a total of 272 ways that they can fill the positions with two different members. How many people are in the Math Club?

Step-by-Step Solution

Answer

The number of the people in the Math Club is 17.

Step 1. Given Information.

Given, the members of the Math Club need to elect a president and a vice-president. They determine that there are a total of 272 ways that they can fill the positions with two different members. It is to be determined how many people are in the Math Club.

Step 2. Calculation .

Let there be n people in the Math Club.

So, electing the president can happen in n ways and then there are n-1 people left for vice-president. So, it can happen in n-1 ways.

By the Fundamental Counting Principle, there are $n\left(n-1\right)$ different ways to elect president and vice-president.

But given there are 272 ways.

Hence,

 $\begin{array}{l}n\left(n-1\right)=272\\ n^2-n=272\\ n^2-n-272=0\\ n^2-17n+16n-272=0\\ n\left(n-17\right)+16\left(n-17\right)=0\\ \left(n-17\right)\left(n+16\right)=0\\ n=17\text{ or }n=-16\end{array}$

Since n represents number of people, it cannot be negative.

Step 3. Conclusion .

Hence, the number of the people in the Math Club is 17.