Find the probability that if the letters of the word "parallel" are randomly arranged that the L's will not be together.

Question

In class, I'm studying permutations and combinations. The solutions stated the no. of permutations where 3 L's are together is 6!/2!. Could you please explain why?

Thanks in advance

Answer 1

Treat the LLL as if they were one element, say X
so we have
X a a p r e, 6 elements of which two are alike, the two a's
number of ways to arrange them is 6!/2!
(remember we divide by 2! because of the two "alikes" )

So in the first part, the number of ways to arrange the original is
8!/(3!2!) = 3360
The number of ways the LLL is together is 6!/2! = 360

Number of ways the L's are NOT together = 3360 - 360 = 3000