(a) There are 6 points in a plane. How many triangles can be formed with the points?
(b) A family of 6 is to be seated in a row . In how many ways can this be done if the father and mother are not to be seated together?
(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.
\(\therefore \text{Father and mother not together =} 6! - 2!5! = 720 - 240 = 480 \)
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