Question

At a certain high school, the Prom Committee is going to choose new members. There are 7 students from the Junior class and 6 students from the Senior class who are willing to be new members. In how many ways can 4 new members be chosen if 2 or fewer must be from the Senior class?

Answer 1

The Solution:

Given:

7 students from the Junior class.

6 students from the Senior class.

4 new members are to be chosen.

Required:

Find the number of ways 4 new members can be chosen if 2 or fewer must be from the senior class.

So, the possibilities are:

`\begin{gathered} (^6C_0\cdot^7C_4)\text{ or }(^6C_1\cdot^7C_3)\text{ or }(^6C_2\cdot^7C_2) \\ \\ (^6C_0\cdot^7C_4)+(^6C_1\cdot^7C_3)+(^6C_2\cdot^7C_2) \end{gathered}`

By formula, Combination is

So,

`\lbrack(6!)/((6-0)!0!)*(7!)/((7-4)!4!)\rbrack+\lbrack(6!)/((6-1)!1!)*(7!)/((7-3)!3!)\rbrack+\lbrack(6!)/((6-2)!2!)*(7!)/((7-2)!2!)\rbrack`
`=(1*35)+(6*35)+(15*21)=35+210+315=560\text{ ways}`

Therefore, the correct answer is 560 ways.