A letter lock consists of three rings marked with 15 different letters. If N denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock then, (A) 48 divides N (B) N is the product of 3 distinct prime numbers(C) N is the product of 4 distinct prime numbers (D) None of these.

To solve this problem, let's go step by step.

Step 1: Understand the lock mechanism
The lock consists of 3 rings.
Each ring has 15 different letters, and each ring can be set to any of those 15 letters.
A "successful" attempt is the correct combination, so all the incorrect combinations are "unsuccessful attempts."
Step 2: Find the total number of possible combinations
Since each ring has 15 letters, and there are 3 rings, the total number of combinations (including the correct one) is:

Total combinations = 15 * 15 * 15 = 15^3 = 3375.

Step 3: Subtract the correct combination
Out of these 3375 combinations, only one combination is correct, so the number of unsuccessful attempts (N) is:

N = 3375 - 1 = 3374.

Step 4: Factorize 3374
We need to check the prime factorization of 3374.

Start by dividing 3374 by the smallest primes:

3374 is an even number, so divide by 2: 3374 ÷ 2 = 1687.
Now, check if 1687 is divisible by smaller primes (3, 5, 7, etc.). 1687 ÷ 7 = 241.
241 is a prime number.
Thus, the prime factorization of 3374 is:

3374 = 2 * 7 * 241.

Step 5: Interpret the result
The prime factorization of 3374 consists of 3 distinct prime numbers: 2, 7, and 241.

Step 6: Answer the question
The correct answer is: (B) N is the product of 3 distinct prime numbers.