Question

The letters of the word LOGARITHM are arranged at random. Find the probability that Exactly 4 letters between G and H.

Answer 1

There are 9 letters in the word LOGARITHM.
These letters can be arranged among themselves in ⁹P₉ = 9! ways.
∴ n(S) = 9!
Let C be the event that exactly 4 letters are arranged between G and H.
Consider the following arrangement
1 2 3 4 5 6 7 8 9
∴ Out of 9 places, G and H can occupy any one of the following 4 positions in 4 ways.
1^st and 6^th, 2^nd and 7^th, 3^rd and 8^th, 4^th and 9^th
Now, G and H can be arranged among themselves in ²P₂ = 2! = 2 ways.
Also, the remaining 7 letters can be arranged in the remaining 7 places in ⁷P₇ = 7! ways.
∴ n(C) = 4 × 2 × 7! = 8 × 7! = 8!

∴ P(C) = `("n"("C"))/("n"("S")`

= `(8!)/(9!)`

= `(8!)/(9xx8!)`

= `1/9`