Comprehensive study guide for CBSE Class 11 Applied Mathematics on linear and circular seating arrangements, covering logical reasoning, constraint analysis, and combinatorial counting methods.
Linear arrangements involve placing $n$ distinct objects in a single row. The number of ways to arrange $n$ objects is given by the factorial of $n$. When specific conditions are introduced—such as individuals sitting together or restricted positions—we treat the 'together' group as a single entity and then multiply by the internal permutations of that group.
• Treat the two specific students as one single unit (a block).
• This leaves us with 4 entities (the block + 3 other students).
• The 4 entities can be arranged in 4! = 24 ways.
• The two students inside the block can switch places in 2! = 2 ways.
• Total arrangements = 24 * 2 = 48.
Answer: There are 48 ways to arrange the students under the given condition.
Circular arrangements differ from linear ones because there is no fixed 'start' or 'end' position. Fixing one position to break the symmetry, the number of ways to arrange $n$ distinct objects in a circle is $(n-1)!$. In cases where clockwise and anti-clockwise arrangements are considered identical (like a necklace or a garland), the formula is divided by 2.
• Identify n = 6.
• Apply the circular permutation formula (n-1)!.
• Calculate (6-1)! = 5!.
• 5! = 5 * 4 * 3 * 2 * 1 = 120.
Answer: There are 120 distinct ways to seat 6 people around a round table.