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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.

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Last updated date: 13th Jun 2024
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Answer
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Hint: We will have to determine no. of samples of 3 rings with 15 different letters first. Then find no. of time someone successfully open the lock and no. of time someone unsuccessfully to open the lock. Finally, we will determine the numerical feature of N accordingly as asked.

Complete step-by-step answer:
Let’s assume a lock which have three rings but 2 letters so,
$\left( {AAA} \right),\left( {AAB} \right)$
 $
  \left( {ABA} \right),\left( {ABB} \right) \\
  \left( {BAA} \right),\left( {BAB} \right) \\
  \left( {BBA} \right),\left( {BBB} \right) \\
 $
These are total 8 i.e. ${2^3}$.
Thus, we may apply this logic to determine the no. of samples of 3 rings with 15 letters lock and then look at the options.
No. of samples of three rings with 15 different letters = ${15^3} = 3375$
But, out of all these permutations only one pattern will be the correct one.
$\therefore $ No. unsuccessful attempts will be (3375-1) = 3374
Now, by factorization of 3374 we will get,
$3374 = 2 \times 7 \times 241$
So, its Prime factors are 2, 7, 241.
$\therefore $ N is the product of 3 distinct prime numbers.

So, the correct answer is “Option B”.

Note: We need to understand the concepts about probability as well as the permutations and combinations. Also, the concept of events and arrangement making will have an important role for solving such problems. Number systems and their various rules are very much applicable for such solutions