Question

A florist has six different colors of roses and three different types of vases. A customer wants to buy four roses and two vases. In how many different ways can that costumer buy three different colored roses and two different types of vases?

Answer 1

Given:

No. of roses having different colours = 6

No. of vases = 3

No. of roses the customer wants = 4

No. of vases the customer wants = 2

To find:

The no. of ways the customer can buy three different coloured roses and two different types of vases.

Solution:

Out of the six different coloured roses, the customer needs three different coloured roses.

No. of ways to buy three different coloured roses from the six different coloured roses = $6C_{3}=\frac{6!}{3!*3!}=\frac{6*5*4*3!}{3*2!*3!}=20$

A customer wants to buy four roses out of which he needs three different coloured roses and one rose to be the same colour.

No. of ways to include that one rose to have the same colour

= $3C_{1}=\frac{3!}{1!*2!} =\frac{3*2!}{1*2!}=3$

The customer needs two different types of vases of the three vases that are available.

No. of ways to buy 2 vases out of the 3 vases = $3C_{2}=\frac{3!}{2!*1!}=\frac{3*2!}{2!*1}=3$

Total no. of ways = $6C_{3}*3C_{1}*3C_{2}=20*3*3=180$ ways

Hence, there are 180 ways to buy three different coloured roses and two different types of vases.

There are 180 ways to buy three different coloured roses and two different types of vases.