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

Question:

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

Solution:

We have n different things.

We are to select 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.

$\therefore$ Required number of ways $={ }^{n-2} C_{r-2} \times(r-1) ! \times 2 !$

$=2(r-1)^{n-2} C_{r-2} \times(r-2) !$

$=2(r-1)^{n-2} P_{r-2}$