The numbers \(1, 2, 3, \ldots, m\) are arranged in random order. The number of ways this can be done, so that the numbers \(1, 2, \ldots, r \, (r < m)\) appear as neighbours is:
Show Hint
- \((m - r)!\)
- \((m - r + 1)!\)
- \((m - r)!r!\)
- \((m - r + 1)!r!\)
The Correct Option is B
Solution and Explanation
Step 1: We are given the numbers \( 1, 2, 3, \ldots, m \), and we need to arrange them such that the numbers \( 1, 2, \ldots, r \) appear as neighbours.
Step 2: Treat the numbers \( 1, 2, \ldots, r \) as a single block. By doing this, we reduce the problem to arranging \( m - r + 1 \) objects: the block and the remaining \( m - r \) numbers.
Step 3: The number of ways to arrange these \( m - r + 1 \) objects is:
\[ (m - r + 1)! \]
Step 4: Within the block, the \( r \) numbers can be arranged in \( r! \) different ways.
Step 5: Therefore, the total number of arrangements is the product of the two:
\[ (m - r + 1)! \times r! \]
Conclusion:
The total number of ways the numbers can be arranged with the numbers \( 1, 2, \ldots, r \) as neighbours is:
\[ (m - r + 1)! r! \]
Top Questions on Probability
- If $ A $ and $ B $ are two events such that $ P(A) = 0.7 $, $ P(B) = 0.4 $ and $ P\left( A \cap \overline{B} \right) = 0.5 $, where $\overline{B}$ denotes the complement of $ B $, then $ P\left( B | \left( A \cup \overline{B} \right) \right) $ is equal to
- JEE Main - 2025
- Mathematics
- Probability
- If the probability that the random variable X takes the value x is given by \( P(X = x) = k(x + 1)3^{-x} \), \( x = 0, 1, 2, 3, ... \), where k is a constant, then \( P(X \ge 3) \) is equal to
- JEE Main - 2025
- Mathematics
- Probability
- All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $ n $ be denoted by $ W_n $. Let the probability $ P(W_n) $ of choosing the word $ W_n $ satisfy $ P(W_n) = 2P(W_{n-1}) $, $ n>1 $. If $ P(CDBEA) = \frac{2^\alpha}{2^\beta - 1} $, $ \alpha, \beta \in \mathbb{N} $, then $ \alpha + \beta $ is equal to :
- JEE Main - 2025
- Mathematics
- Probability
- A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is:
- If \( P(A) = \frac{1}{5}, P(B) = \frac{3}{5} \) and \( P(A \cap B) = \frac{2}{5} \), then \( P(A' \cup B') \) is :
Questions Asked in WBJEE exam
- The variation of displacement with time of a simple harmonic motion (SHM) for a particle of mass \( m \) is represented by: \[ y = 2 \sin \left( \frac{\pi}{2} + \phi \right) \, \text{cm} \] The maximum acceleration of the particle is:
- WBJEE - 2025
- simple harmonic motion
- Ruma reached the metro station and found that the escalator was not working. She walked up the stationary escalator with velocity \( v_1 \) in time \( t_1 \). On another day, if she remains stationary on the escalator moving with velocity \( v_2 \), the escalator takes her up in time \( t_2 \). The time taken by her to walk up with velocity \( v_1 \) on the moving escalator will be:
- WBJEE - 2025
- Relative Motion
- A force \( \mathbf{F} = ai + bj + ck \) is acting on a body of mass \( m \). The body was initially at rest at the origin. The co-ordinates of the body after time \( t \) will be:
- WBJEE - 2025
- Newtons Laws of Motion
A quantity \( X \) is given by: \[ X = \frac{\epsilon_0 L \Delta V}{\Delta t} \] where:
- \( \epsilon_0 \) is the permittivity of free space,
- \( L \) is the length,
- \( \Delta V \) is the potential difference,
- \( \Delta t \) is the time interval.
The dimension of \( X \) is the same as that of:- WBJEE - 2025
- Dimensional Analysis
- Which logic gate is represented by the following combination of logic gates?
- WBJEE - 2025
- Logic gates