Question

In how many ways can four girls and two boys be arranged in a row if a) the boys are on each end of the row? b) the boys must be together? c) the boys must be together in the middle of the row?

Short answer

a) 48, b) 240, c) 48

Step-by-step solution

Define the Problem for Part (a)

We need to arrange four girls and two boys in a row such that the boys are on each end of the row.

Place the Boys in Specific Positions

The boys must be on each end, so the possible arrangement for boys is fixed as B _ _ _ _ B, where B represents a boy.

Determine the Number of Ways to Arrange the Boys

There are 2 boys, which can be arranged in 2! ways at the ends of the row. Hence, the number of ways to arrange the boys is 2! = 2.

Arrange the Girls

The 4 girls can be arranged in the remaining 4 positions in 4! ways. Thus, the number of ways to arrange the girls is 4! = 24.

Calculate Total Arrangements for Part (a)

Multiply the number of ways to arrange the boys by the number of ways to arrange the girls: 2! * 4!. So, the total number of arrangements is 2 * 24 = 48.

Define the Problem for Part (b)

We need to arrange four girls and two boys in a row with the condition that the boys must be together.

Treat the Boys as a Single Unit

Consider the two boys as a single unit. So, we essentially have 5 units to arrange: (BB), G, G, G, G.

Arrange the Five Units

The 5 units can be arranged in 5! ways. Thus, the number of ways to arrange the units is 5! = 120.

Arrange the Boys Within Their Unit

The 2 boys within their unit can be arranged in 2! ways. Thus, the number of ways to arrange the boys within their unit is 2! = 2.

Calculate Total Arrangements for Part (b)

Multiply the number of ways to arrange the units by the number of ways to arrange the boys within their unit: 5! * 2!. So, the total number of arrangements is 120 * 2 = 240.

Define the Problem for Part (c)

We need to arrange four girls and two boys in a row with the boys together in the middle of the row.

Place the Boys Together in the Middle

Since the boys must be in the middle, the arrangement looks like: _ _ BB _ _. There are 4 positions available for the girls.

Determine the Number of Ways to Arrange the Boys

The 2 boys together in the middle can be arranged in 2! ways. Thus, the number of ways to arrange the boys is 2! = 2.

Arrange the Girls

The 4 girls can be arranged in the 4 remaining positions in 4! ways. Thus, the number of ways to arrange the girls is 4! = 24.

Calculate Total Arrangements for Part (c)

Multiply the number of ways to arrange the boys by the number of ways to arrange the girls: 2! * 4!. So, the total number of arrangements is 2 * 24 = 48.

Key concepts

Factorials

Factorials are a fundamental concept in combinatorics and mathematics at large. The notation for a factorial is represented by an exclamation point (!) after a number. It means the product of all positive integers up to that number. For example, \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
Factorials are used in permutations and combinations to count how many ways you can arrange or choose items. Understanding how to calculate and use factorials is essential for solving combinatorial problems. You can see in the example exercise how we used factorials to determine different arrangements for boys and girls.

Arranging People

Arranging people, or permutations, is about determining the number of ways to organize individuals in a specific order. In the given exercise, we dealt with arranging boys and girls in a row under varying conditions.

• In part (a): The boys were fixed at both ends, leaving room to arrange the girls in the middle.
• In part (b): The boys were considered a single unit (treated as one block) that could move around together.
• In part (c): The boys needed to be in the middle positions, fixing part of the arrangement automatically.

Remember, any specific condition given in the problem alters how you must approach the arrangement.

Combinatorial Problems

Combinatorial problems involve finding the number of ways to arrange or select items. It often combines both permutations (arrangements) and combinations (selections).

• Arrangements (permutations) take order into account, making the sequence important.
• Selections (combinations) do not consider the order, only the chosen elements matter.

The exercise provided illustrates a combinatorial problem where specific constraints influence our counting principles. By following each condition step-by-step, we can tackle complex arrangements efficiently.

Counting Principles

Counting principles, like the fundamental counting principle, help solve combinatorial problems by breaking them down into manageable parts. The basic rule states that if there are \( n \) ways of doing one thing and \( m \) ways of doing another, there are \( n \times m \) ways of doing both.
In the exercise:

• For part (a): We multiplied the ways to arrange boys by the ways to arrange girls (\( 2! \times 4! \)).
• For part (b): We treated the boys as a single unit, calculated the arrangements of the units, then multiplied by the arrangements within the boy unit (\( 5! \times 2! \)).
• For part (c): The boys in the middle were arranged first, followed by the girls (\( 2! \times 4! \)).

Using these principles makes evaluating different scenarios systematic and straightforward.