Question

How many different words can be formed with the word CUSTOM with a condition that the word should begin with M?

• A. 720
• B. 540
• C. 120
• D. 180

Answer 1

The Correct Option is C

To determine the number of different words that can be formed with the word "CUSTOM" under the condition that the word should begin with the letter "M", follow these steps:

1. Fix the starting letter: Since the word must begin with "M", the first letter is fixed as "M".
2. Count the remaining letters: After fixing "M" as the first letter, we have the letters "C", "U", "S", "T", and "O" remaining.
3. Calculate permutations: The number of permutations of these 5 letters is given by the factorial of the number of letters. That is 5 letters can be arranged in 5! (5 factorial) ways.
4. Compute 5!: 5! = 5 × 4 × 3 × 2 × 1 = 120.

Thus, the number of different words that can be formed is 120.

Answer 2

CUSTOM has 6 different letters, so we can also solve this by looking at the total arrangements and using symmetry instead of arranging the leftover letters directly.

1. Option A (720): This is the count of all arrangements of the 6 letters with no restriction at all, so it does not answer the question asked.
2. Option B (540): There is no natural count in this problem that lands on 540, so this does not fit.
3. Option C (120): Out of all 720 arrangements, M is equally likely to sit in any of the 6 positions, so the count with M first is \( 720 \div 6 = 120 \).
4. Option D (180): There is no natural count in this problem that lands on 180 either, so this does not fit.

The count of 720 total arrangements split evenly across the 6 possible first letters gives 120 arrangements with M first, matching option C.

Let's summarize:

• All 6 letters give \( 6! = 720 \) total arrangements.
• By symmetry, each letter is first in exactly one-sixth of them.

Therefore, the correct answer is C (120).