Find the number of permutations of n distinct things taken r together,

1.	Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.
Answer» 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 – 3}C_{r – 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 – 3}C_{r – 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 – 3}C_{r – 3} × 3! × (r – 2)!

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

Answer»

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 – 3}C_{r – 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 – 3}C_{r – 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 – 3}C_{r – 3} × 3! × (r – 2)!

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

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