Question

8 chairs are numbered 1 to 8 Two women and 3 men wish to occupy one chair each first the women choose the chairs marked, 1 to 4 then the men select from among the remaining chairs. The number of seating arrangements is (a) 1250 (b) 1360 (c) 1440 (d) 1560

Short answer

The total number of seating arrangements is 288.

Step-by-step solution

Calculate the number of arrangements for the women.

Since there are two women and they have to choose from 4 chairs, we can use the permutation formula to find the number of seating arrangements for them: Number of arrangements = nPr where n is the total number of chairs and r is the number of women. Here, n = 4 and r = 2. Using the formula, we have: Number of arrangements for women = 4P2 = \(\frac{4!}{(4-2)!}\) = \(\frac{4!}{2!}\)

Calculate the factorial values for the women arrangements.

In this step, we calculate the factorial values for both the numerator and the denominator: 4! = 4 × 3 × 2 × 1 = 24 2! = 2 × 1 = 2

Divide the factorial values to get the number of arrangements for the women.

Now, we divide the factorial values to find the arrangements for the women: Number of arrangements for women = \(\frac{24}{2}\) = 12 So, there are 12 possible seating arrangements for the two women among the 4 chairs.

Calculate the number of arrangements for the men.

Now, we need to find the number of seating arrangements for the three men among the remaining 4 chairs (since there are 8 chairs in total). We can use the same permutation formula: Number of arrangements for men = 4P3 = \(\frac{4!}{(4-3)!}\) = \(\frac{4!}{1!}\)

Calculate the factorial values for the men arrangements.

In this step, we calculate the factorial values for both the numerator and the denominator: 4! = 4 × 3 × 2 × 1 = 24 1! = 1

Divide the factorial values to get the number of arrangements for the men.

Now, we divide the factorial values to find the arrangements for the men: Number of arrangements for men = \(\frac{24}{1}\) = 24 So, there are 24 possible seating arrangements for the three men among the 4 remaining chairs.

Find the total number of seating arrangements.

Lastly, to find the total number of seating arrangements possible, we multiply the number of arrangements for the women and the men: Total number of seating arrangements = Number of arrangements for women × Number of arrangements for men Total number of seating arrangements = 12 × 24 = 288 The correct option is not given in the exercise, but the total number of seating arrangements is 288.

Key concepts

Factorial

A factorial, denoted by an exclamation mark (!), is a mathematical operation that multiplies a series of descending natural numbers. It is crucial when calculating the permutations or arrangements in combinatorial problems.
For a given number \( n \), the factorial \( n! \) is calculated as \( n \times (n-1) \times (n-2) \times \, ... \, \times 1 \).
Some key points about factorials are:

• Factorial is only applicable to non-negative integers.
• 0! is defined as 1, because it is the empty product.
• Factorials grow very quickly with larger numbers.

In our exercise, factorials help us determine permutations by calculating the total possible arrangements of a subset of items. For example, \( 4! \) was used to find potential seating options for both women and men.

Seating Arrangements

Seating arrangements involve the process of arranging individuals or objects in a specific order. It is a practical application of permutations, which is the calculation of different possible orders.
In scenarios with multiple choices and restrictions, like specific chairs for men and women, the problem becomes a combinatorial challenge.
Some important considerations include:

• Identifying which seats are available for each group.
• Using permutations to find the possible arrangements for each group.

In the provided problem, the two women initially choose their seats from 4 options, followed by the three men selecting from the remaining seats. This step-by-step process maximizes the total potential arrangements.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of elements within a set. It heavily relies on concepts such as permutations and combinations to solve problems related to configurations and structures. Combinatorics can apply to anything from organizing seating to solving complex probability queries.
Three main ideas in combinatorics include:

• Permutations: Arranging items where the order matters.
• Combinations: Selecting items where order does not matter.
• Factorials: As used to calculate permutations and combinations.

In our exercise, combinatorics helps break down the problem into smaller steps. We use permutations (seating women, then men) to ensure all possible configurations are calculated and accounted for.