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# Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

Given that we need to find the no. of permutations formed by r things which were taken from n distinct things in which 3 particular things must occur together.

Here, it is clear that 3 things are already selected and we need to choose (r – 3) things from the remaining (n – 3) things.

Let us find the no. of ways of choosing (r – 3) things and assume it to be N1.

N1 = (No. of ways of choosing (r – 3) things from remaining (n – 3) things)

N1 = n – 3Cr – 3

Now we need to find the no. of permutations than can be formed using 3 things which are together.

Now we need to arrange the chosen 3 things. Since every thing differs from other. The arrangement is similar to that of arranging n people in n places which are n! ways to arrange. So, the total no. of words that can be formed is 3!.

Now let us assume the together things as a single thing this gives us total (r – 2) things which were present now.

Now, we need to arrange these (r – 2) things. Since every thing differs from other. The arrangement is similar to that of arranging n people in n places which are n! ways to arrange. So, the total no. of words that can be formed is (r – 2)!.

Let us the total no. of words formed be N.

N = N1 × 3! × (r – 2)!

N = n – 3Cr – 3 × 3! × (r – 2)!

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)!.

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