A letter lock consists of three rings with 15 | vedclass

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(C) The total number of possible combinations for the three rings is $15^3 = 3375$.There is only $1$ correct combination to open the lock.Therefore, t

Vedclass · Accepted Answer

$N$ is the product of three distinct prime numbers.

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(C) The total number of possible combinations for the three rings is $15^3 = 3375$.There is only $1$ correct combination to open the lock.Therefore, the number of unsuccessful attempts $N$ is given by $N = 15^3 - 1$.Using the algebraic identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, we have $N = (15 - 1)(15^2 + 15 	imes 1 + 1^2) = 14 	imes (225 + 15 + 1) = 14 	imes 241$.Since $14 = 2 	imes 7$, we get $N = 2 	imes 7 	imes 241$.Here, $2$, $7$, and $241$ are all prime numbers.Thus, $N$ is the product of three distinct prime numbers.

$A$ letter lock consists of three rings with $15$ different letters each. If $N$ denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then:

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