(a) There are 6 points in a plane. How many triangles can be formed with

Explanation

(a) Number of triangles that can be formed = \(^{6}C_{3}\)

= \(\frac{6!}{(6 - 3)! 3!} = \frac{6 \times 5 \times 4}{3 \times 2}\)

= 20 ways.

(b) Tie the father and mother together so that there are 5 persons to arrange in 5! ways.

Changing the positions of the father and mother, this can be done in 2! ways.

\(\therefore \text{Ways of arranging the family = } 2!5!\)

Without restriction, they can be arranged in 6! ways.

\(\therefore \text{Father and mother not together =} 6! - 2!5! = 720 - 240 = 480 \)