Question

Given below are two statements :
A number of distinct 8-letter words are possible using the letters of the word SYLLABUS. If a word in chosen at random, then
Statement I: The probability that the word contains the two S's together is \(\frac{1}{4}.\)
Statement II : The probability that the word begins and ends with L is \(\frac{1}{28}.\)
In the light of the above statements, choose the correct answer from the options given below.

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is A

To solve this problem, we need to determine if both statements about probabilities in forming words from the letters of "SYLLABUS" are correct.

1. Calculate the total number of distinct 8-letter words formed from "SYLLABUS".

   Letters in "SYLLABUS": S, Y, L, L, A, B, U, S.

   Since 'L' and 'S' are repeated, the total number of arrangements is given by:

   \(\frac{8!}{2! \times 2!}\)

   Calculating this:

   \(\frac{40320}{4} = 10080\)

2. Check Statement I: "The probability that the word contains the two S's together is \(\frac{1}{4}\)."

   Treat the two 'S's as a single unit. Thus, we have 7 units to arrange: 'SS', Y, L, L, A, B, U.

   The number of ways to arrange these units is:

   \(\frac{7!}{2!}\)

   Calculating this:

   \(\frac{5040}{2} = 2520\)

   The probability is then:

   \(\frac{2520}{10080} = \frac{1}{4}\)

   Thus, Statement I is correct.

3. Check Statement II: "The probability that the word begins and ends with L is \(\frac{1}{28}\)."

   Fix 'L' at both ends: L _ _ _ _ _ _ L. Now, we arrange the remaining: S, S, Y, A, B, U (6 unique letters).

   The number of ways to arrange these letters is:

   \(\frac{6!}{2!}\)

   Calculating this:

   \(\frac{720}{2} = 360\)

   The probability is then:

   \(\frac{360}{10080} = \frac{1}{28}\)

   Thus, Statement II is correct.

Conclusion: Both Statement I and Statement II are true, so the correct answer is "Both Statement I and Statement II are true".