Question

To furnish a room in a model home, an interior decorator is to select 2 chairs and 2 tables from a collection of chairs and tables in a warehouse that are all different from each other. If there are 5 chairs in the warehouse and if 150 different combinations are possible, how many tables are in the warehouse?A. 6
B. 8
C. 10
D. 15
E. 30

Answer 1

Answer:

A. 6

Step-by-step explanation:

Let n represent number of tables.

We have been given that to furnish a room in a model home, an interior decorator is to select 2 chairs and 2 tables from a collection of chairs and tables in a warehouse that are all different from each other. There are 5 chairs in the warehouse and if 150 different combinations are possible.

Since 2 chairs are being selected from 5 chairs, so we can choose 2 chairs in $5C2$ ways.

There are n tables and we can choose 2 tables from n table in $nC2$ ways.

We can represent our given information in an equation as:

$5C2\times nC2=150$

$\frac{5!}{2!(5-2)!}\times \frac{n!}{2!(n-2)!}=150$

$\frac{5*4*3!}{2*1*3!}\times \frac{n!}{2!(n-2)!}=150$

$10\times \frac{n!}{2!(n-2)!}=150$

$\frac{n!}{2!(n-2)!}=15$

$\frac{n!}{2*1*(n-2)!}=15$

$\frac{n*(n-1)*(n-2)!}{2*(n-2)!}=15$

$\frac{n(n-1)}{2}=15$

$n(n-1)=30$

$n^2-n=30$

$n^2-n-30=0$

$n^2-6n+5n-30=0$

$n(n-6)+5(n-6)=0$

$(n-6)(n+5)=0$

$(n-6)=0\text{ (or) }(n+5)=0$

$n=6\text{ (or) }n=-5$

Since tables cannot be negative quantity, therefore, 6 tables are in the warehouse.