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प्रश्न
The letters of the word LOGARITHM are arranged at random. Find the probability that Exactly 4 letters between G and H.
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उत्तर
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.
1st and 6th, 2nd and 7th, 3rd and 8th, 4th and 9th
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`
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