(Q). In a school, roll no. of k students are given from 1 to k. Three students are selected from these k students such that their roll numbers are not consecutive. In how many ways selection can be made?
Solution:-
roll numbers are:-
1, 2, 3, 4, 5, 6, 7, …………….., (k-2), (k-1), (k)
1ˢᵗ consider the number of ways when all the three roll number are consecutive. Then possible selections can be:-
(1, 2, 3) (2, 3, 4) (3, 4, 5) (4, 5, 6)…………….((k-2) (k-1)(k))
above selection of 3 consecutive roll numbers can be done in (k-2) ways
& number of ways of selection of three when roll number when there is no restriction = ᵏC₃
thus required number of ways = number of selection without any restriction – number of selection with restriction
= ᵏC₃ – (k-2)
= – (k-2)
= – (k-2)
=
= Answer