permutations and combinations!

[ss_xo]

how many permutations are there of the letters of the word NONILLION
In how many of these permutations are the N's separated??

 

[soroban]

Hello, ss_xo!

How many permutations are there of the letters of the word NONILLION?

There are 9 letters, but there are: 2 I's, 2 L's, 3 N's, and 2 O's.

There are: .\(\displaystyle \dfrac{9!}{2!\,2!\,3!\,2!} \:=\:7,\!560\) permutations.

In how many of these permutations are the N's separated?

Place the I's, L's, and O's in a row.
Insert spaces before, after and between them.
. . \(\displaystyle \_\,I\,\_\,I\,\_\,L\,\_\,L\,\_\,O\,\_\,O\,\_\)

There are: .\(\displaystyle \dfrac{6!}{2!\,2!\,2!} = 90\) ways to arrange the six letters.

Select 3 of the 7 spaces and insert the N's.
. . There are: .\(\displaystyle {7\choose3} \:=\:\dfrac{7!}{3!\,4!} \:=\:35\) ways to insert the N's.


Therefore, there are: .\(\displaystyle 90\times 35 \:=\:3,\!150\) permutations
. . . in which there are no adjacent N's.**