If the letters of the word REGULATION' be arranged at random, the probability that there will be exactly 4 letters between R and E isa)1/10b)1/9c)1/5d)1/2Correct answer is option 'B'. Can you explain this answer?

Question

Answer

To find the probability that there will be exactly 4 letters between R and E in the word REGULATION, we need to determine the total number of possible arrangements and the number of favorable arrangements.

Total Number of Possible Arrangements:
The word REGULATION has a total of 10 letters. Therefore, there are 10! (10 factorial) ways to arrange these letters without any restrictions.

Number of Favorable Arrangements:
To have exactly 4 letters between R and E, we can consider R and E as a single entity. So, we have the arrangement: (RE _ _ _ _ _ _ _ _ _ ), where the underscores represent the remaining 8 letters.

Now, we can arrange these 9 entities (RE and the remaining 8 letters) in 9! ways. However, within the entity (RE), there are 2 letters that can be arranged in 2! ways. Therefore, the number of favorable arrangements is 9! * 2!.

Calculating the Probability:
The probability of an event is given by the formula:
Probability = Number of favorable outcomes / Total number of possible outcomes

Therefore, the probability that there will be exactly 4 letters between R and E in the word REGULATION is:
Probability = (9! * 2!) / 10!

Simplifying the expression:
9! / 10! = 1/10

Therefore, the correct answer is option B) 1/9.

Answer

R E G U L A T IO N
It has 10 letters. Four letters are there between R and E, if there positions are
1, 6; 2, 7; 3, 8; 4, 9 or 5, 10 i.e. 5 ways
So, required probability is

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