In how many ways can the letters of the word ‘PARALLEL’

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 

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