Question

In how many ways can 15 members of a school sit along a circular table, when the secretary is to sit on side of the principal and the deputy secretary on the other side ? (a) \(2 \times 12 !\) (b) 24 (c) \(2 \times 15 !\) (d) \(2 ! \times 13 !\)

Short answer

The total number of ways the 15 members can be arranged along the circular table is \(2 \times 11!\), where we treat the principal, secretary, and deputy secretary as a single group. This takes into account the two possible internal arrangements of these three positions. The correct answer is \(2 \times 11!\).

Step-by-step solution

Create a Group

First treat the principal, secretary, and deputy secretary as a single group. We can call this the "principal group." Now we have a total of 13 people (as the "principal group" is counted as one entity) to arrange along the circular table.

Find the Number of Arrangements

Since the "principal group" is now just one person in terms of arrangement, there are 12 people to be arranged along with this group. Remember that for a circular arrangement, there are (n-1)! ways to arrange n people. So, we have 12 people to arrange, which will be done in (12-1)! = 11! ways.

Consider Internal Arrangements

Now, we need to consider the internal arrangement of the principal, secretary, and deputy secretary within the "principal group." There are 2 possible arrangements: secretary-principal-deputy secretary, or deputy secretary-principal-secretary. So, we have 2 ways to arrange them.

Calculate the Total Number of Arrangements

To find the total number of ways the 15 members can be arranged along the circular table, we multiply the number of arrangements for the 12 people plus the "principal group" arrangement (11!) by the number of possible internal arrangements (2). This gives us: Total number of arrangements = 2 × 11! So, the correct answer is (a) 2 × 11!.

Key concepts

Permutations

A permutation refers to the arrangement of objects in a specific order. For example, consider arranging 3 books on a shelf: book A, book B, and book C. The possible permutations are ABC, ACB, BAC, BCA, CAB, and CBA.The number of permutations of n distinct objects is given by the formula:\[ n! \]where \(!\) signifies factorial, meaning you multiply all positive integers up to \(n\). For instance, \(3! = 3 \times 2 \times 1 = 6\).In the context of our exercise, permutations help determine how many ways we can arrange people around a table. For linear seating, all seats are different, and every permutation counts as a unique arrangement.

• Linear permutations consider every different position.
• Factorials grow quickly as \(n\) increases.
• Used for determining orders and sequences.

Combinatorial Arrangements

Combinatorial arrangements involve counting and arranging objects based on a set of rules or constraints. These arrangements go beyond simple permutations by considering conditions that might limit the order or selection of objects. In our exercise, not only are we arranging people, but we also have specific roles located next to each other, forming a group. By treating this group as a single entity, we effectively reduce the total number of items to arrange. Some examples of constraints in combinatorial arrangements include:

• Grouping specific elements together, as in the exercise (principal, secretary, and deputy secretary).
• Allowing some items to be interchangeable.
• Fixing certain items in specific positions.

These arrangements are crucial in determining valid permutations from cases that include particular restrictions.

Circular Seating Arrangements

Circular seating arrangements are a specialized type of permutation.In these setups, we arrange objects along a closed loop, such as a round table. One unique challenge with circular arrangements is that rotations that differ only by starting point are considered identical. Thus, for \(n\) items in a circle, the number of distinct arrangements is given by:\[ (n-1)! \]The reason is simple: fix one position to determine a unique starting point, permuting the rest.In the problem exercise, once we treat the principal, secretary, and deputy secretary as one group, they contribute to the "fixed" point. The remaining 12 people, including this group, are then arranged in a circle:

• Only \((n-1)!\) arrangements are considered due to rotational symmetry.
• Suitable for puzzles and games that require circular scenarios.
• Often include constraints like adjacency requirements.

Understanding circular arrangements aids in practical applications such as designing seating plans and organizing rotations.