Question

If three boys and three girls sit in a row on a park bench, and no boy can sit on either end of the bench, how many arrangements of the children on the bench are possible? a. 46,656 b. 38,880 c. 1,256 d. 144 e. 38

Short answer

There are 144 possible arrangements, meaning the correct choice is (d) 144.

Step-by-step solution

Positioning the Girls

Since no boy can sit on either end of the bench, it's a given that girls will be positioned at both ends. In addition, one girl will be in the middle. Therefore, there are 4 possible spaces where the girls and boys can sit: _ G _, _ G _, _ G _. The girls can be arranged in these three chosen spots in 3! (3-factorial) ways.

Positioning the Boys

With the girls positioned, there are essentially 4 slots remaining where the boys can sit, designated by the underscores in the girl arrangement. These are equivalent to the spaces between the girls and at the ends of the bench. The boys can be arranged in these four spots in 4P3 (Permutation of 4 slots taken 3 at a time) ways.

Calculating the total arrangements

The total number of arrangements is obtained by multiplying the number of ways the girls can arrange and the number of ways the boys once the girls have been seated. Thus it is 3! * 4P3.

Numerical computation

Calculate 3!(=6) * 4P3 [=4!/(4-3)!] (=24), which equals to 144.

Key concepts

Factorial

One of the foundational concepts in permutations and combinations is the 'factorial'. The factorial function, denoted by an exclamation mark (!), is the product of all positive integers up to a given number. For example, the factorial of 3, written as 3!, is calculated as follows:

• 3! = 3 × 2 × 1 = 6

Factorials are crucial for finding permutations and combinations. They help us determine the number of ways we can arrange a set number of items. In our exercise, the arrangement of the girls on the bench requires us to compute 3!, as there are 3 girls that can be organized in different seating orders. Understanding the concept of factorial is essential as it lays the groundwork for more complex mathematical problems in combinatorial mathematics.

Permutation

Permutations are arrangements of objects in a specific order. The order really matters in permutations. If you're asked to arrange a set of items where the sequence is important, you're dealing with permutations. In this mathematical problem at hand, once the positions for the girls are established, the boys must be seated in the remaining slots. The permutation formula, denoted as nPr, helps compute this.

• nPr = \( \frac{n!}{(n-r)!} \)

Here, n is the total number of spots and r is the number of items to arrange. In our exercise, we must place 3 boys into 4 available slots, computed as 4P3. This formula ensures every possible configuration of boys is considered.

Combinatorial Problems

Combinatorial problems involve figuring out the number of ways things can be combined or arranged. These problems are foundational in mathematics, often appearing in fields like statistics, computer science, and more. In our bench arrangement problem, we're dealing with a classic combinatorial challenge. The key steps are:

• Determining restrictions (boys can't sit at the ends).
• Fixing certain positions (girls at the ends).
• Calculating permutations for the remaining elements (boys in this instance).

The goal in such problems is to count all feasible arrangements meeting the criteria. This requires an understanding of both permutations and factorial layers to arrive at the correct solution.

Mathematics Problem Solving

Mathematics problem solving is more than just plugging numbers into formulas. It involves a set of skills to approach and solve complex problems systematically. Breaking down the bench problem involves a step-by-step method. Steps we used include:

• Understanding the problem constraints (where certain gendered students can sit).
• Identifying the sequence of operations (arranging girls first, then boys).
• Performing calculations correctly (using factorials and permutation formulae).

This structured approach isn't just useful for combinatorial issues but for myriad mathematical scenarios. By analyzing, planning, computing, and verifying, students improve not only their math skills but also their logical reasoning and critical thinking capabilities.