Question

A home owner wants to purchase two different chairs for the living room, one for in front of the fire place and one for in front of the bay window. An interior decorator arrives at the home with photos of several different chairs, and shows the owner all 72 possible arrangements. Photos of how many different chairs were shown to the home owner?

Answer 1

Using the combination formula, it is found that photos of 9 different chairs were shown to the home owner.

The order in which the chairs are chosen is important, hence, the permutation formula is used to solve this question.

What is the permutation formula?

$P_{n,x}$ is the number of different permutations of x objects from a set of n elements, given by:

$P_{n,x} = \frac{n!}{(n-x)!}$

In this problem, two chairs are chosen from a set of n, and there are 72 different outcomes, hence:

$P_{n,x} = 72, x = 2$

$P_{n,x} = \frac{n!}{(n-x)!}$

$72 = \frac{n!}{(n-2)!}$

$72 = \frac{n(n - 1)(n - 2)!}{(n-2)!}$

$n^2 - n = 72$

$n^2 - n - 72 = 0$

Which is a quadratic function with coefficients $a = 1, b = -1, c = -72$, hence:

$\Delta = b^2 - 4ac = (-1)^2 - 4(1)(-72) = 289$

Taking the positive root:

$x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{1 + \sqrt{289}}{2} = 9$

Photos of 9 different chairs were shown to the home owner.

You can learn more about the permutation formula at brainly.com/question/25925367