Question

The number of permutations of n distinct objects taken r at a time in which three particular objects occurs together is ___________.

Answer 1

Total number of objects (distinct) = n
Since, 3 objects are taken together (i.e 3 objects are together)
The number of ways of selecting r distinct objects from n is ⁿC_r
If three objects occur together, then number of ways = ⁿ⁻³C_r−3
also number of arrangements of these three things = 3!
and number of arrangements of (r − 3 + 1) objects = (r − 2)!
∴ Total possible ways of taking r at a time from n objects such that three particular are together is ⁿ⁻³C_r−3 (r − 2)! 3!