In how many ways can the letters of the following word taken two at a time, be selected?

Question

In how many ways can the letters of the following word taken two at a time, be selected? 'MONSOON'

In the word 'MONSOON', there are $3 O's$ and $2 N's$. How could the selection be Made?

Answer 1

There are four unique letters here: $M,O,N,S$.

So the number of ways of choosing two distinct letters at a time is ${4 \choose 2} = 6$. We can then add the duplicate pairs which are $O,O$ and $N,N$ to give a total of $8$ possible pairs.

If we care about order (but without distinguishing between the three $O$s and two $N$s) then we have $4! = 24$ possible pairs of distinct letters, to which we once again add $2$ more to give 26 possible pairs with order.