Question

Evaluate the combinations. $$ C_{3}^{5} $$

Short answer

Question: Evaluate the combinations formula \(C_{5}^{3}\) using the provided values. Answer: Using the combinations formula: \(C_{5}^{3} = \frac{5!}{3!(5-3)!}\), we find that \(C_{5}^{3} = 10\).

Step-by-step solution

Identify values of n and r

For this exercise, we have \(n = 5\) and \(r = 3\). Our goal is now to compute \(C_{5}^{3}\) using the combinations formula.

Compute the factorials

Before we plug the values into the formula, we need to compute the factorials for \(n\), \(r\), and \((n-r)\). Factorials are the product of all the positive integers up to that number, such as: \(n! = 1*2*3*...*(n-1)*n\). In our case, we need to compute \(5!\), \(3!\), and \((5-3)!\). $$ 5! = 5 * 4 * 3 * 2 * 1 = 120 \\ 3! = 3 * 2 * 1 = 6 \\ (5-3)! = 2! = 2 * 1 = 2 $$

Use the combinations formula

We substitute the values of \(n = 5\) and \(r = 3\) into the combinations formula: $$ C_{5}^{3} = \frac{5!}{3!(5-3)!} = \frac{120}{6 * 2} $$

Calculate the result

Now, we simply need to perform the mathematical operations to find the combinations: $$ C_{5}^{3} = \frac{120}{6 * 2} = \frac{120}{12} = 10 $$ Hence, the value of \(C_{5}^{3}\) is 10.

Key concepts

Factorials

Factorials are fundamental to understanding problems that involve permutations and combinations. The concept of a factorial is relatively simple: it is the product of all positive integers up to a given number. To denote the factorial of a number, we use the exclamation point symbol. For instance, the factorial of 5, written as \(5!\), is calculated by multiplying all positive integers from 1 up to 5 together:

\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

A key point to remember is that the factorial of zero is defined as one, \(0! = 1\). This is not immediately intuitive, but it is a convention that simplifies many mathematical expressions. Understanding factorials is crucial because they are part of the foundation of many probability formulas, and they are essential when calculating permutations and combinations which are based on factorial operations. A strong grasp of how to compute factorials will make it much easier to deal with more complex probability problems.

Permutations and Combinations

Permutations and combinations are two concepts in probability that deal with the arrangement of objects. While they seem similar, they address different scenarios.

Permutations

Permutations are concerned with arrangements where the order matters. For example, the number of ways to arrange a set of books on a shelf where the sequence is important would be a permutation problem.

Combinations

Conversely, combinations pertain to scenarios where the order does not matter. If you are interested in the number of ways to choose a committee from a group of people, and the order in which they are chosen is irrelevant, you have a combination problem at hand. The formula to calculate 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 being chosen, and \(n - r\) represents the difference between the two. By mastering these concepts and understanding when to use each, students can handle a wide array of problems that involve selecting or arranging different sets of items.

Probability Formulas

Probability formulas are the tools that help us quantify the likelihood of events. They are based on a ratio: the number of favorable outcomes over the total number of possible outcomes. Understanding the basic premises of probability is vital for solving problems that ask for the chance of a specific event occurring.

When using permutations and combinations within probability calculations, we often analyze scenarios such as lotteries, card games, or even genetics. For example, to find the probability of drawing a particular hand in poker, we'd use combinations to calculate the total number of possible hands. Then to find the probability, we'd use the formula:

\(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\).

For the chosen hand, the number of favorable outcomes would be determined by the specific hand's combination, and the total number of possible outcomes would be the combination of all possible hands. Working through these formulas step by step allows students to dissect complex problems into more manageable pieces which can be crucial in ensuring that they don't just get the right answer but understand the process that led them there.