Question

Find the number of permutations of n different things taken r at a time such that two specified things occur together?

Answer 1

We have n different things.
We are to select r things at a time such that two specified things occur together.
Remaining things = n - 2 

Out of the remaining (n - 2) things, we can select (r - 2) things in ^{n- 2C_{r -2}}ways.

Consider the two things as one and mix them with (r - 2) things.

Now, we have (r - 1) things that can be arranged in (r - 1)! ways.

But, two things can be put together in 2! ways.

\[\therefore \text{Required number of ways} = {}^{n - 2} C_{r - 2} \times \left( r - 1 \right)! \times 2!\]
\[ = 2 \left( r - 1 \right)^{n - 2} C_{r - 2} \times \left( r - 2 \right)!\]
\[ = 2 \left( r - 1 \right)^{n - 2} P_{r - 2}\]