Question

How many ways can you arrange all the letters and numbers in "KCET 2025" which start with K and end with 5?

• A. 720
• B. 360
• C. 120
• D. 180

Hint:

Always carefully scan the string for repeated characters. Forgetting to divide by the factorial of the count of identical items is the most common error in these types of permutation problems.

Answer 1

The Correct Option is B

Step 1: Identify total characters
The characters in ``KCET 2025'' (ignoring space) are: \[ \{K, C, E, T, 2, 0, 2, 5\} \] Total characters $= 8$
Step 2: Apply given conditions
First position is fixed as $K$ and last position is fixed as $5$: \[ [K] \ _ \ _ \ _ \ _ \ _ \ _ \ [5] \] Remaining positions $= 6$
Step 3: Remaining characters
\[ \{C, E, T, 2, 0, 2\} \] Here, digit $2$ is repeated twice.
Step 4: Use permutation formula
Number of distinct arrangements: \[ \frac{6!}{2!} \]
Step 5: Calculate
\[ = \frac{720}{2} = 360 \] Final Answer:
\[ \boxed{360} \]