To solve the problem of how many ways we can arrange all the letters and numbers in "KCET 2025" such that they start with 'K' and end with '5', we can follow these steps:
Fix the First and Last Characters: Since the arrangement must start with 'K' and end with '5', we can fix these two positions. This leaves us with the characters 'C', 'E', 'T', '2', '0', '2', which need to be arranged in the middle five positions.
Count the Remaining Characters: There are 6 characters left to arrange: 'C', 'E', 'T', '2', '0', '2'. Notice that '2' appears twice.
Apply the Permutation Formula: To find the number of ways to arrange 6 items with repetition, use the permutation formula:
\(\frac{n!}{p_1! \cdot p_2! \cdot \ldots \cdot p_k!}\)
Where \(n\)is the total number of items, and \(p_1, p_2, \ldots, p_k\)are the frequencies of the repeated items.
Here, \(n = 6\)(total characters), and the character '2' repeats twice.
Plug these values into the formula:
\(\frac{6!}{2!} = \frac{720}{2} = 360\)
Conclusion: Therefore, there are 360 ways to arrange "KCET 2025" with the specific condition that the string starts with 'K' and ends with '5'.
The correct answer is therefore 360.