Out of 6 teachers and 8 students, a committee of 11 is being formed. In how many ways can this be done, if the committee contains

(i) exactly 4 teachers?

(ii) at least 4 teachers?

Since the committee of 11 is to be formed from 6 teachers and 8 students.

(i) Forming a committee with exactly 4 teachers

Choosing 4 teachers out of 6 in ⁶C₄ ways.

Remaining 7 from 8 students in ⁸C₇ ways.

Thus, total ways in (i) are ⁶C₄ ⁸C₇ ways.

(ii) The number of ways in this case is

1. 4 teachers and 7 students

2. 5 teachers and 6 students

3. 6 teachers and 5 students

= ⁶C₄⁸C₇+⁶C₅⁸C₆⁶C₆⁸C₅

Applying ⁿC_r =

= 344 ways