To find the number of permutations of the word PARALLEL in which the 3 L's do not come together, we can use the concept of permutations with restrictions.

1. Calculate the total number of permutations:
The word PARALLEL has 8 letters in total. Therefore, the total number of permutations without any restrictions would be 8!.

2. Calculate the number of permutations when the 3 L's are together:
Consider the 3 L's as a single entity. So, we have 6 distinct entities: P, A, R, A, E, and LL. The number of permutations of these 6 entities is 6!.

3. Subtract the number of permutations when the 3 L's are together from the total number of permutations:
Total number of permutations - Number of permutations with 3 L's together = 8! - 6!

4. Calculate the number of permutations when 3 L's are together in a single entity:
Now, consider the 3 L's as a single entity. So, we have 6 distinct entities: P, A, R, A, E, and LLL. The number of permutations of these 6 entities is 6!.

5. Calculate the number of permutations when the 3 L's are together in any order:
Since the 3 L's can be arranged in any order within the entity, we need to consider the number of permutations of the 3 L's themselves. The number of permutations of 3 L's is 3!.

6. Subtract the number of permutations when the 3 L's are together in any order from the number of permutations when the 3 L's are together:
Number of permutations with 3 L's together - Number of permutations with 3 L's together in any order = 6! - 3!

7. Calculate the final answer:
Total number of permutations - Number of permutations with 3 L's together + Number of permutations with 3 L's together in any order = 8! - 6! + 3!

Simplifying this expression:
8! - 6! + 3! = 40320 - 720 + 6 = 39606

Therefore, the number of permutations of the word PARALLEL in which the 3 L's do not come together is 39606. However, none of the given options match this answer. Hence, the correct answer cannot be determined from the given options.

The word "PARALLEL" can be arranged in

8!÷ 2!×3! ways which is 3360

Now find the No. of the ways in which all the L's come together is

Total no. of L = 3
considering all the L as a single digit, add it to the remaining letters which 5+1 = 6
which is

6! ÷ 2! = 360

The no. of ways in which 3 Ls do not come together =
total no. of ways the word can be arranged - the no of ways 3 L's come together.

3360 - 360 = 3000

Hope this helps ...
Thank you