Question:
In how many ways can the letters of the word ‘PARALLEL’ be arranged so that all L’s do not come together?
Solution:
To find: number of words where L do not come together
Let the three L’s be treated as a single letter say Z
Number of words with L not the together = Total number of words - Words with L’s together
The new word is PARAEZ
Total number of words $=\frac{8 !}{2 ! 3 !}=3360$
Words with $L$ together $=6 !=720$
⇒ Words with L, not together = 3360 – 720 = 2640
There are 2640 words where L do not come together