Problem 47 - Algebra for College Students

Question

In how many ways can six people be seated at a circular table? [Hint: Moving each person one place to the right (or left) does not create a new seating.] 120

Step-by-Step Solution

Answer

120 ways.

Step 1: Understanding Circular Permutation

When arranging people in a circle, one position is typically fixed to avoid counting equivalent rotations as different arrangements. This effectively reduces the problem from a permutation of \( n \) people to a permutation of \( n-1 \) people.

Step 2: Calculate Permutations

With six people, fixing one person's seat leaves us with 5 people to arrange. The number of permutations of 5 people is calculated as \(5!\).

Step 3: Apply Permutation Formula

The formula for permutations of \( n \) distinct items is \( n! = n \times (n-1) \times ... \times 1\). Here, we compute \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Step 4: Conclude the Problem

Since we've found \(5!\) equals 120, there are 120 distinct ways to arrange 6 people around a circular table.

Key Concepts

Factorial

A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a specified number. It plays a crucial role in permutations, combinations, and other areas of mathematics. The factorial helps calculate the number of ways to arrange or sequence items.

Consider factorials as products:

• The factorial of 1, written as 1!, is simply 1.
• The factorial of 2, written as 2!, is 2 times 1, which equals 2.
• Progressing further, the factorial of 3, written as 3!, equals 3 × 2 × 1 = 6.
• Continue this pattern, for example, 4! = 4 × 3 × 2 × 1 = 24, and so on.

When considering permutations of elements in a set, factorials facilitate computations, particularly as the number of elements increases. It is vital to use factorials appropriately, especially in problems involving arrangements where factorial growth is rapid.

To simplify understanding, try factorial calculations by hand to experience the exponential growth as numbers increase. It’s a foundational tool in mathematics, laying a base for more complex concepts.

Permutations

Permutations refer to the different ways of arranging a set of items where the order is important. When dealing with circular permutations, the arrangement forms a circle, introducing a unique challenge. In circular arrangements, shifting all elements together doesn't create a new permutation, which reduces the count of distinct arrangements.

The basic permutation formula is:\[n! = n \times (n-1) \times (n-2) \times \ldots \times 1\]To calculate permutations in a circle, such as arranging people around a table, we fix one person in position. This adjustment handles the rotational symmetry where rotating everyone around doesn’t change the arrangement. Thus, instead of calculating for all individuals, calculate for one less:

• For a group of six, fix one person, then arrange the remaining five (calculate 5!).
• This results in \(5!=120\) unique ways.

Understanding this adjustment is fundamental to solving circular permutation problems correctly.

Combinatorics

Combinatorics is a fascinating branch of mathematics concerned with counting, arrangement, and combination of objects. It integrates various principles like permutations, combinations, and the factorial notation to solve complex counting problems efficiently.

In the context of seating people around a table, combinatorics simplifies determining exact arrangements through permutations. It reveals not just the methodology, but the logic behind variations in arrangements, distinguishing between linear and circular arrangements.

• Linear arrangements consider a straight line with no repetition of equivalent sequences.
• Circular arrangements, as discussed, require fixing a point to adjust for rotations.

Combinatorics essentially let's us calculate - the number of ways objects can be organized, - considering constraints like position and order.

It empowers problem-solvers to break down complex problems into digestible steps and find solutions that would otherwise seem daunting. Dive into combinatorics to not just solve textbook problems, but to gain insights into how different arrangements & combinations can fundamentally alter an organization.