Problem 2 - Thinking Mathematically

Question

Three men and three women line up at a checkout counter in a store. a. In how many ways can they line up? b. In how many ways can they line up if the first person in line is a woman, and then the line alternates by genderthat is a woman, a man, a woman, a man, and so on? c. Find the probability that the first person in line is a woman and the line alternates by gender.

Step-by-Step Solution

Answer

a. There are 720 ways to line up. b. There are 36 ways to alternate by gender with a woman coming first. c. The probability that the first person in line is a woman and the line alternates by gender is 1/20.

Step 1: Compute total arrangements

There are 6 people total, hence they can line up in \(6!\) ways which is 720.

Step 2: Compute arrangements where a woman is first and genders alternate

For every alternate position starting from the first one being a woman, we can arrange 3 women and 3 men each in \(3!\) ways. Hence total such arrangements are \(3! \times 3!\) which is 36.

Step 3: Compute probability of an alternate arrangement with woman being first

The probability is calculated by dividing the number of the desired outcome by the total number of outcomes. Hence, the probability is \(\frac{36}{720} = \frac{1}{20}\)

Key Concepts

Permutations

Permutations are a fascinating concept in combinatorics. They refer to the different ways in which a set or number of items can be arranged in order. In our exercise, the task was to determine how many ways three men and three women can line up. The key point here is the order, because the same group of people in a different sequence forms a different permutation.
For example, even if you have the same six people, rearranging them results in a distinct permutation. The general formula for permutations of n items is given by the factorial of n, denoted as \(n!\). This represents multiplying all positive integers up to n.

• A simpler way to visualize it is: for 6 people, the number of permutations is \(6!\), which equals 720.
• This includes every different possible order of the individuals.

PERMUTATION FORMULA - Given any set of items, say n, the total number of permutations is \(n!\). This shows how to calculate the different possible orderings of the group. For alternating sequences such as having alternating genders in the arrangement, additional permutation formulas are applied. Permutations make sure that all possible sequences are calculated thoroughly.

Probability

Probability is a measure of how likely an event is to occur. It's an essential concept in statistics, helping us gauge the chances of a specific outcome. In the given exercise, the problem of probability arises in question c, where we need to determine the chances that the first person in line is a woman, and the genders alternate thereafter.
Understanding probability involves dividing the number of successful outcomes by the total number of possible outcomes.

• In our exercise, there is exactly one situation where this gender alternation happens with a woman at the start, giving 36 successful arrangements.
• The total possible ways the group can line up is 720.
• Hence, the probability is calculated as \( \frac{36}{720} \), which simplifies to \( \frac{1}{20} \).

This means there is a 5% chance of having a woman first and alternating genders.

Factorials

The factorial of a number is a concept that appears frequently in permutations and combinations. Denoted by \(!\), it signifies the product of all natural numbers up to a specific number. In mathematical terms, \(n! = n \times (n - 1) \times (n - 2) \times \ldots \times 1\). Factorials are crucial for determining the number of ways objects can be arranged or selected.
In our exercise, factorials come into play when calculating arrangements for the queue.

• For the total number of ways 6 people can line up, the factorial \(6!\) is used, resulting in 720.
• For alternating gender arrangements, \(3!\) was used for both the men and women to find permutations, producing 36 possible arrangements.

LEVERAGING FACTORIALS - They give us a quick formula to calculate large sets of permutations, making complex problems manageable by simplifying computation.