If the letters of word 'ATTRACTION' are written down at random then which of the following statement are TRUE:
(A) Probability that all the T's occurs together is \( \displaystyle\frac{1}{15} \)
(B) Probability that all the T's are together is \( \displaystyle\frac{7}{15} \)
(C) Probability that no two T's occurs together is \( \displaystyle\frac{7}{15} \)
(D) Probability that no two T's occurs together is \( \displaystyle\frac{3}{15} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
ATTRACTION has 10 letters, with A twice and T three times. The total number of arrangements is
\( \displaystyle \frac{10!}{2!\,3!}=302400 \)
All the T's together. Treat TTT as one block. Then there are 8 objects with A repeated:
\( \displaystyle \frac{8!}{2!}=20160 \)
\( \displaystyle P=\frac{20160}{302400}=\frac{1}{15} \)
No two T's together. Arrange the other 7 letters first. They leave 8 gaps, and the 3 T's go into 3 different gaps:
\( \displaystyle \frac{7!}{2!}=2520 \)
\( {}^{8}C_{3}=56 \)
\( \displaystyle P=\frac{2520\times 56}{302400}=\frac{7}{15} \)
So P(all T's together) \( =\frac{1}{15} \) and P(no two T's together) \( =\frac{7}{15} \).
Checking the statements
(A) Probability that all the T's occurs together is \( \displaystyle\frac{1}{15} \) — Correct. The value is \( \frac{1}{15} \).
(B) Probability that all the T's are together is \( \displaystyle\frac{7}{15} \) — Incorrect. The value is \( \frac{1}{15} \).
(C) Probability that no two T's occurs together is \( \displaystyle\frac{7}{15} \) — Correct. The value is \( \frac{7}{15} \).
(D) Probability that no two T's occurs together is \( \displaystyle\frac{3}{15} \) — Incorrect. The value is \( \frac{7}{15} \).