Algebra 2 help Independent and Dependent Events?

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? Please show me how to work it out. Thanks a bunch.

Answer 1

Electing 2 people from the Math Club for different positions is conceptually the same as sitting 2 people from the Math Club in 2 different chairs.  Thus, this is a permutation problem.  Let x=number of people in the Math Club.  Then the number of 2-element permutations from an x element set is equal to 272.  Define mPn=m!/(m-n)!, where mPn gives the number of n-element permutations possible from an m-element set.  Then xP2=272.

Then,

x!/(x-2)!=272

x(x-1)(x-2)!/(x-2)!=272

x(x-1)=272

x^2-x=272

x^2-x-272=0

x=17.

So, there are 17 people in the Math Club.