(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?
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 )