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 $-$2}C_{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.

$$\begin{gathered} \therefore \mathrm{Required}\mathrm{number}\mathrm{of}\mathrm{ways}={}^{n-2}{C}_{r-2}\times \left(r-1\right)!\times 2! \\ =2{\left(r-1\right)}^{n-2}{C}_{r-2}\times \left(r-2\right)! \\ =2{\left(r-1\right)}^{n-2}{P}_{r-2} \end{gathered}$$