Problem 20 - Algebra for College Students

Question

Four girls and three boys are to be randomly seated in a row of seven seats. Find the probability that the girls and boys will be seated in alternating seats. \(\frac{1}{35}\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{35}\).

Step 1: Determine Total Arrangements

First, we calculate the total number of ways to arrange seven individuals (four girls and three boys) in a row. The total number of permutations for these seven individuals is 7!, which is 5040.

Step 2: Identify Alternating Seat Pattern Possibilities

Next, consider alternating seats for boys and girls. If a girl starts in the first seat, the pattern is G-B-G-B-G-B-G. Alternatively, if a boy starts, the pattern would be B-G-B-G-B-G-B.

Step 3: Calculate Arrangements for Pattern 1 (G-B-G-B-G-B-G)

For the pattern G-B-G-B-G-B-G, the four girls can fill the G positions in 4! (24) ways, and the three boys can fill the B positions in 3! (6) ways.

Step 4: Calculate Arrangements for Pattern 2 (B-G-B-G-B-G-B)

Similarly, for the pattern B-G-B-G-B-G-B, the three boys can fill the B positions in 3! (6) ways, and the four girls can fill the G positions in 4! (24) ways.

Step 5: Combine Possibilities

Add the number of arrangements from both alternating patterns: (24 * 6) + (6 * 24) = 144 + 144 = 288.

Step 6: Calculate Probability

Finally, divide the number of successful arrangements by the total number of arrangements to find the probability. The probability is \(\frac{288}{5040}\). Simplifying gives \(\frac{1}{35}\).

Key Concepts

Alternating Patterns

When considering how to seat four girls and three boys in a way that genders alternate, it's essential to think about alternating patterns. You can start imagining these patterns like unique sequences. Whether the first seat is occupied by a girl or a boy results in different alternating patterns.
For instance, if the first seat is taken by a girl, the pattern proceeds as G-B-G-B-G-B-G. Conversely, if a boy starts, the pattern alternates as B-G-B-G-B-G-B. These distinct sequences are completely separate paths which ensure that no two boys or two girls sit next to each other. Understanding these patterns allows us to systematically arrange the seating in a way that meets the exercise's requirements.

Factorial Calculations

Factorials are a mathematical concept that calculates the number of ways to arrange a set number of items. In this exercise, we use factorial calculations to determine permutations. For example, to seat seven people in a row, we calculate the factorial of seven, denoted as 7!, which equals 5040.
Factorials efficiently establish how many different sequences can be formed from a group of distinct items. For example, 4! (24) shows how many ways the four girls can be arranged when choosing seats specifically for them, similar to 3! (6) for the boys. Understanding how to calculate these can simplify solving probability exercises that involve arrangements and permutations.

Successful Arrangements

Successful arrangements are the specific setups that meet the conditions set out in a problem. Here, an arrangement is successful if boys and girls alternate starting from either gender. Once the alternating patterns are understood, we find out the number of arrangements for each pattern:

• For G-B-G-B-G-B-G, there are 4! ways to arrange the girls and 3! for the boys, totaling 24 * 6.
• For B-G-B-G-B-G-B, it reverses, so again 3! * 4! sequences to try, also 24 * 6.

Both successful patterns contribute 144 arrangements each, summing up to 288 successful arrangements.Finally, to find the probability of these arrangements out of all possible ones, you divide the number of successful arrangements by total arrangements. In this exercise, we have \(\frac{288}{5040}\), simplifying down to \(\frac{1}{35}\). This simplification process provides an easy-to-understand result of how likely it is to seat the children in the alternating order as required.