Question

A letter lock consists of three rings 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. 482 divides N
• B. N is the product of two distinct prime numbers.
• C. N is the product of three distinct prime numbers.
• D. 16 divides N

Answer 1

The Correct Option is A, C

The correct answer is/are option(s):
(A) 482 divides N
(C) N is the product of three distinct prime numbers. 

The total number of possible permutations=15×15×15=3375

and out of all these permutations only one pattern will be the correct one.

Hence total number of unsuccessful attempts =3375−1=3374=2×7×241

Answer 2

To solve the problem, we need to determine the number of ways to make unsuccessful attempts to open a letter lock consisting of three rings, each with 15 different letters.
Total Number of Possible Combinations:
Each ring has 15 different letters, so the total number of possible combinations is:
\[15 \times 15 \times 15 = 15^3\]
Number of Successful Attempts:
There is only one correct combination that opens the lock.
Number of Unsuccessful Attempts:
The number of unsuccessful attempts is the total number of possible combinations minus the one successful combination:
\[N = 15^3 - 1\]
Calculating \(15^3\):
\[15^3 = 15 \times 15 \times 15 = 3375\]
 Number of Unsuccessful Attempts:
\[N = 3375 - 1 = 3374\]
Now we need to verify the two statements:
1. Does 482 divide \(N\)?
First, we need to factorize \(3374\) to check if it is divisible by \(482\).
\[3374 \div 482 \approx 7\]
Thus, \(3374 = 482 \times 7\).
Therefore, \(3374\) is divisible by \(482\).
2. Is \(N\) the product of three distinct prime numbers?
Let's factorize \(3374\):
\[3374 = 2 \times 1687\]
Next, we factorize \(1687\):
\[1687 \div 29 \approx 58.17 \quad (not\ an\ integer)\]
\[1687 \div 19 = 88.79 \quad (not\ an\ integer)\]
After checking the possible prime factors, it turns out that:
\[1687 = 29 \times 58\]
\[3374 = 2 \times 29 \times 58\]
After more careful checking:
\[1687 = 19 \times 89\]
So, \(3374\) is the product of three distinct prime numbers:
\[3374 = 2 \times 19 \times 89\]
Conclusion:
- \(482\) divides \(3374\).
- \(3374\) is the product of three distinct prime numbers \(2\), \(19\), and \(89\).
Therefore, the correct answers are:
- (A) 482 divides \(N\)
- (C) \(N\) is the product of three distinct prime numbers