Question

The 10 letters of the word STATISTICS are arranged randomly. The probability that the 10-letter arrangement starts with CAT is

• A. \(\frac{1}{80}\)
• B. \(\frac{1}{120}\)
• C. \(\frac{1}{240}\)
• D. \(\frac{1}{280}\)
• E. \(\frac{1}{360}\)

Hint:

To simplify probability calculations with factorials, always cancel the common terms (like \(7!\) and \(3!\)) before multiplying out large numbers. It reduces the chance of arithmetic error.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Probability = (Ways letters start with CAT) (Total ways to arrange letters).

Step 2: Key Formula or Approach:
Arrangement of \(n\) items with repeats: \(\frac{n!}{p! q! r! \dots}\).
Word "STATISTICS": S-3, T-3, A-1, I-2, C-1. Total = 10.

Step 3: Detailed Explanation:
1. Total arrangements: \[ \text{Total} = \frac{10!}{3! 3! 2!} \]
2. Favorable arrangements (starting with CAT): The first three positions are fixed as C, A, T. Remaining letters to arrange: S-3, T-2 (one T used), I-2. Total = 7. \[ \text{Favorable} = \frac{7!}{3! 2! 2!} \]
3. Probability \(P\): \[ P = \frac{7! / (3! 2! 2!)}{10! / (3! 3! 2!)} = \frac{7!}{3! 2! 2!} \times \frac{3! 3! 2!}{10!} \] \[ P = \frac{7!}{2!} \times \frac{3!}{10!} = \frac{7! \times 6}{2 \times 10 \times 9 \times 8 \times 7!} \] \[ P = \frac{3}{10 \times 9 \times 8} = \frac{3}{720} = \frac{1}{240} \]

Step 4: Final Answer:
The probability is \(\frac{1}{240}\).