Problem 36 - Calculus for Biology and Medicine

Question

Paula, Cindy, Gloria, and Jenny have dinner at a round table. In how many ways can they sit around the table if Cindy wants to sit to the left of Paula?

Step-by-Step Solution

Answer

There are 4 ways.

Step 1: Understand the Problem

We need to find the number of distinct arrangements for four people around a round table where Cindy must sit to the left of Paula. A round-table arrangement requires considering rotational symmetries.

Step 2: Count Total Arrangements

For seating four people around a round table, we fix one person's position due to rotation symmetry and arrange the others. Thus, there are \[(4 - 1)! = 3! = 6\] ways to seat 4 people in a round table.

Step 3: Apply the Restriction

Cindy must sit to the left of Paula. Once Paula's position is fixed, Cindy's position is also fixed relative to Paula's, reducing the problem to arranging the remaining two people (Gloria and Jenny).

Step 4: Calculate Arrangements with Restriction

Since Cindy's position is fixed relative to Paula, treat Cindy and Paula as a block with fixed orientation. Now, we have to arrange this block and two other people. This can be done in \[2! = 2\] different ways.

Step 5: Combine Results

Combining Steps 3 and 4, we find the number of valid ways to seat everyone. Multiply the arrangements counting the Cindy-Paula restriction by the arrangement of the rest:\[2 \times 2 = 4\] different ways.

Key Concepts

Permutations

Permutations are a fundamental concept in combinatorics that help us determine the number of ways to arrange a set of items. They are particularly useful when we need to account for the sequence in which items appear. In this exercise involving Paula, Cindy, Gloria, and Jenny sitting around a round table, permutations are central to finding the solution. Since the seating is around a table, we deal with circular permutations. When arranging four people in a line, there are typically 4! permutations. However, in a circular arrangement, one position is fixed to account for identical rotations. Hence, only (4-1)! = 3! = 6 permutations need to be considered.

• A linear permutation accounts for each item having a distinct position in sequence.
• A circular permutation reduces redundancy by fixing one position, due to the same arrangements being possible by simply rotating the circle.

Rotational Symmetry

Rotational symmetry is a geometric concept that applies whenever an object looks the same after a certain amount of rotation. It is a key factor in solving puzzles or problems involving round tables or circular objects.In this problem, seating arrangements around a circular table exhibit rotational symmetry. For example, if you arrange the four girls in a sequence around the table, and then rotate the table, the sequence looks the same but has shifted positions. This means several arrangements would appear identical if we didn't account for this rotation. To simplify counting, we generally fix one person's position (as done in Step 2) which systematically eliminates symmetrically equivalent arrangements. This is why the formula for circular permutations is \[(n-1)! \]instead of the standard linear permutation equation \(n!\). This understanding helps reduce the problem space significantly.

Constraint-Based Arrangement

In many permutation scenarios, certain conditions or constraints must be satisfied, such as specific relative positions between items. Here, Cindy must be to the left of Paula. Constraints like this require modifications to standard permutation formulas. In our current problem, Cindy's required position relative to Paula reduces possibilities more than usual. We treat Cindy and Paula as a single unit or block, focusing only on the internal arrangement of this block.

• Identify and fix constraints: Here it's Cindy's specific placement relative to Paula.
• Create blocks for constrained items: Treat Cindy and Paula as a single unit.
• Reduce remaining permutations: From three available positions, we now consider two free arrangements for the block, simplifying further to a 2! calculation for the block and remaining individuals.

By applying these constraints correctly, we reduced the number of valid permutations, ensuring each requirement of the problem was met.