Combination Question

[oxvt]

I can't figure this one out.

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

[BuckeyeT]

The total number of combinations should be...
# of chair combinations x # of table combinations = total # of combinations

Since we can calculate the # of chair combinations (5 to choose from, pick 2), and we know the total # of combinations (150), we can determine the # of table combinations (and # of tables to choose from).

Let TC2 = # of table combinations,
5C2 x TC2 = 150
(5! / ((5-2)! 2!)) x TC2 = 150
(5 x 4)/2 x TC2 = 150
10 x TC2 = 150
TC2 = 15

At this point, I just plugged in a value for T (# of tables to choose from). Since TC2 = 15, T should be slightly larger than 5 (since 5C2 = 10).

Let's try 6.

6! / ((6-2)! 2!)
(5 x 6)/ 2
30/2
15

So, the # of tables in the warehouse equals 6.

[rsadana1]

I did it the exact same way. This is a good problem.

I would like to add one small thing. When the equation was written for the total number of combination, we used the Principle of Multiplication and multiplied the number of ways to select chair with the number of ways to select tables. This is because the question requires selection of BOTH chairs and tables. Principle of Multiplication and Addition are two very fundamental principles that help us to formulate the equations to solve these problems.