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
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.
Let \( A_1 \): People with good health,
\( A_2 \): People with average health,
and \( A_3 \): People with poor health.
During a pandemic, the data expressed that the chances of people contracting the disease from category \( A_1, A_2 \) and \( A_3 \) are 25%, 35% and 50%, respectively.
Based upon the above information, answer the following questions:
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