Question

How many words can be formed with the letters of the word 'PARALLEL' so that all L's do not come together?

Answer 1

There are 8 letters in the word 'PARALLEL' out of which A's and 3 are L's and the rest are all distinct.
So, total number of words = $\frac{8!}{2!3!}$
= $\frac{8\times 7\times 6\times 5\times 4\times 3!}{2\times 1\times 3!}$
= $8\times 7\times 6\times 5\times 2$
= 3360
Considering all L's together and treating them as one letter we have 6 letters out of which A repeats 2 times and others are distinct. These 6 letters can be arranged in $\frac{6!}{2!}$ ways.
So, the number of words in which all L's come together = $\frac{6!}{2!}$= $\frac{6\times 5\times 4\times 3\times 2!}{2!}$ = $6\times 5\times 4\times 3=360$
Hence, the number of words in which all do not come together = 3360-360 = 3000