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

A. 6

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* nC2=150`

`(5!)/(2!(5-2)!)* (n!)/(2!(n-2)!)=150`

`(5*4*3!)/(2*1*3!)* (n!)/(2!(n-2)!)=150`

`10* (n!)/(2!(n-2)!)=150`

`(n!)/(2!(n-2)!)=15`

`(n!)/(2*1*(n-2)!)=15`

`(n*(n-1)*(n-2)!)/(2*(n-2)!)=15`

`(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.