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

Updated On: Apr 11, 2025

482 divides N
N is the product of two distinct prime numbers.
N is the product of three distinct prime numbers.
16 divides N

Hide Solution

Verified By Collegedunia

The Correct Option is A, C

Approach Solution - 1

Total possible combinations: Each ring has 15 letters: \[ 15 \times 15 \times 15 = 3375 \] Number of unsuccessful attempts: \[ N = 3375 - 1 = 3374 \] Check (A): Does 482 divide N? Check if: \[ 3374 \div 482 = 7 \Rightarrow 3374 = 482 \times 7 \Rightarrow \text{Yes, 482 divides } N \] Check (C): Is N the product of 3 distinct prime numbers? Prime factorization of 3374: \[ 3374 = 2 \times 1687 \] Factor 1687: \[ 1687 = 19 \times 89 \Rightarrow 3374 = 2 \times 19 \times 89 \] All are distinct primes, so yes.
Correct options: (A) ✅ 482 divides N (C) ✅ N is the product of three distinct primes

Was this answer helpful?

Hide Solution

Verified By Collegedunia

Approach Solution -2

Total permutations: Each digit can be chosen in 15 ways (say, from a set of 15 symbols). So, the total number of possible 3-digit patterns: \[ 15 \times 15 \times 15 = 3375 \] Unsuccessful attempts: Only one is the correct pattern, so: \[ 3375 - 1 = 3374 \] Now factorize 3374: \[ 3374 = 2 \times 7 \times 241 \] Hence: Let \(N = 3374\), then: - Option (A)(C) Correct options: (A) ✅ (C) ✅

Was this answer helpful?

Questions Asked in WBJEE exam

View More Questions

Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points.

A straight line can be represented as an equation in various forms, as show in the image below:

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y₀ = m (x – x₀)

2. Two – Point Form

Let's look at the line. L crosses between two places. P₁(x₁, y₁) and P₂(x₂, y₂) are general points on L, while P (x, y) is a general point on L. As a result, the three points P₁, P₂, and P are collinear, and it becomes

The slope of P₂P = The slope of P₁P₂, i.e.

\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)

Hence, the equation becomes:

y - y₁ =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c