Question

In a committee of 10 students and 5 teachers, how many different groups of five can you send to a conference

a) if there must be only 1 teacher

b) If there must be at least 1 student.

Answer 1

a)

Given

Number of students = 10

Number of teachers = 5

Possible ways of choosing 1 teacher from 5 teachers is  

There are 4 people remaining to make groups of five

Possible ways of choosing 4 students from 10 students is  

So,

Number of possible groups of five that can be send to a conference if there must be only 1 teacher

= 210 * 5

= 1050 ways

b)

Total number of people = 10 + 5 = 15

Number of possible ways of choosing 5 people from 15 people is  

Number of possible ways of choosing all 5 teachers =

Now

Number of different groups of five that can be send to a conference if there must be at least 1 student

=Total number of possible ways of choosing 5 from 15 - number of possible ways of choosing all 5 teachers

=

= 3003 - 1

= 3002 ways