It is clear that 3 things are already selected and we need to choose (r – 3) things from the remaining (n – 3) things.
Lets find the no. of ways of choosing (r – 3) things.
The number of ways = (No. of ways of choosing (r – 3) things from remaining (n – 3) things)
= n – 3Cr – 3
We need to find the no. of permutations than can be formed using 3 things which are together. Therefore, the total no. of words that can be formed is 3!
Let us assume the together things as a single thing this gives us total (r – 2) things which were present now. Therefore, the total no. of words that can be formed is (r – 2)!
The total number of words formed is:
n – 3Cr – 3 × 3! × (r – 2)!
Hence, the no. of permutations that can be formed by r things which are chosen from n things in which 3 things are always together is n – 3Cr – 3 × 3! × (r – 2)!