Question

Tell whether the question can be answered using permutations or combinations. Explain your reasoning. Then answer the question. An employee at a pet store needs to catch 5 tetras in an aquarium containing 27 tetras. In how many groupings can the employee capture 5 tetras?

Short answer

The question can best be answered using combinations as the order in which the tetras are caught doesn't matter. On calculating the number of groupings using combinations, the solution is found by simplifying \( C(27, 5) = \frac{27!}{5!(27-5)!} \).

Step-by-step solution

Understanding permutations and combinations

In permutations, the arrangement or order matters. In combinations, order does not matter. Here, we are looking for groupings or combinations of 5 tetras. Whether tetra A is caught first or tetra B doesn’t affect the overall outcome – that is, a group of 5 fish. So, combinations should be used here.

Combinations formula

The formula for combinations is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \(n\) is the total number of items, \(r\) is the number of items to choose, '!' denotes factorial.

Applying the combinations formula

In this problem, \(n = 27\), as there are 27 tetras and \(r = 5\) as 5 tetras need to be captured. Substituting these values into the formula, the number of groupings in which the employee can capture the tetras will be \( C(27, 5) = \frac{27!}{5!(27-5)!} \).

Simplifying the factorial and expression

Solving the factorial and simplifying the expression would give us the number of groupings. Thus, the final answer would be the solved value of the expression \( C(27, 5) = \frac{27!}{5!(27-5)!} \).

Key concepts

Permutations vs Combinations

When it comes to calculating the number of ways objects can be arranged or selected, two concepts come into play: permutations and combinations.

Permutations refer to the arrangements of items where order is crucial. For instance, the sequence in which runners finish a race could be a question of permutations, because who comes in first, second, and third matters. If we have three runners A, B, and C, the arrangement ABC is different from ACB.

Combinations, on the other hand, are used when the order of selection does not make a difference. If you're creating a bouquet using 3 types of flowers out of 10 types available, it only matters which types you select, not the order they're picked in. Here, the combination of flowers A, B, and C is the same as C, B, and A.

In the textbook exercise, the employee selects 5 tetras out of 27. It does not matter which tetra is caught first, only that five unique tetras are caught. Thus, we use combinations to find the answer.

Factorial Notation

Factorial notation is foundational in combinations and permutations. It's symbolized by an exclamation mark (!) and indicates the product of all positive integers up to a given number.

For example, the factorial of 5, written as \(5!\), is calculated as \(5 \times 4 \times 3 \times 2 \times 1 = 120\). It's important to remember that \(0!\) is defined as 1. Factorials grow very quickly with larger numbers, making them significant in calculating possibilities in various scenarios.

Understanding factorial is crucial when working with the combinations formula, as we'll see next.

Combinations Formula

The formula for combinations allows us to calculate how many different groups can be made from a larger set when the order doesn't matter. The combinations formula is written as:

\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Where \(n\) represents the total number of items to pick from, and \(r\) is the number of items we are selecting.

In the pet store scenario, we are not concerned with the order in which tetras are caught, making the combination formula applicable. The store clerk can create \( C(27, 5) = \frac{27!}{5!(27-5)!} \) unique groupings of 5 tetras from a total of 27, using the combinations formula. This formula simplifies the process of determining possible groupings, regardless of the magnitude of numbers involved.

Probability and Counting

Probability and counting are deeply interconnected. Counting principles, such as combinations and permutations, are often used to determine the likelihood of events occurring.

For instance, if we want to know the probability of catching a specific group of tetras, we'd count the number of favorable combinations and divide it by the total possible combinations. This relationship between counting and probability supports decision-making processes, provides insights into likely outcomes, and predicts events in various fields like genetics, gaming, and risk assessment.