Problem 47 - Algebra and Trigonometry

Question

41–54 These problems involve combinations. Choosing Exam Questions A student must answer seven of the ten questions on an exam. In how many ways can she choose the seven questions?

Step-by-Step Solution

Answer

She can choose the seven questions in 120 ways.

Step 1: Understand the problem

We need to determine how many different ways a student can choose 7 questions out of 10 on an exam. This is a classic combinations problem, as the order in which the questions are chosen doesn't matter.

Step 2: Define the formula for combinations

The formula to determine the number of combinations is given by: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(n\) is the total number of items to choose from, and \(k\) is the number of items we are choosing.

Step 3: Identify the values in the problem

Here, \(n = 10\) (the total number of questions) and \(k = 7\) (the number of questions to be chosen).

Step 4: Plug the values into the formula

Substitute the values into the combinations formula: \( \binom{10}{7} = \frac{10!}{7!(10-7)!} \). Simplifying, this becomes \( \frac{10!}{7!3!} \).

Step 5: Calculate factorials

Calculate the factorials needed: \(10! = 3628800\), \(7! = 5040\), and \(3! = 6\).

Step 6: Compute the combination

Now compute \( \frac{10!}{7!3!} = \frac{3628800}{5040 \times 6} = \frac{3628800}{30240} = 120 \).

Step 7: Conclude the result

Therefore, the student can choose 7 questions out of 10 in 120 different ways.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, particularly in the study of combinations and permutations. A factorial is denoted by an exclamation mark (!) and signifies the product of all positive integers up to a certain number. For example, 5 factorial, or written as \(5!\), is \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
Factorials are crucial for calculating combinations because they help determine how many ways you can arrange a set of objects. The larger the number of objects, the larger the factorial value, which means more possible arrangements or choices.
Here's a quick way to remember some basic factorials:

• \(0! = 1\) by definition, which is a special case.
• \(1! = 1\)
• \(2! = 2\times 1 = 2\)
• \(3! = 3\times 2\times 1 = 6\)
• \(4! = 4\times 3\times 2\times 1 = 24\)

Each of these factorials is a building block for calculating more complex problems involving combinations.

Combinatorial Analysis

Combinatorial analysis is a method used in mathematics to calculate the number of ways certain events can occur. It's a key concept for understanding how elements are selected or arranged. In the context of choosing exam questions, combinatorial analysis helps in calculating the exact number of combinations possible.
The main principle in combinatorial analysis is to focus on selection processes where the order does not matter, which is different from permutations where order does matter. When you're dealing with combinations, you're working within a set number of elements and selecting them in various ways.
Some key points to remember are:

• Identify whether the problem is a combination or permutation.
• Compute the number of elements you're working with (\(n\)).
• Determine how many selections to make (\(k\)).
• Apply the combination formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Through combinatorial analysis, you can explore various possibilities and come to a conclusive answer for how many combinations can be formed.

Binomial Coefficient

The binomial coefficient is a numerical value that represents the number of ways to choose a subset of items from a larger set, and it's typically denoted as \( \binom{n}{k} \). This coefficient is central to the study of combinations because it directly answers the question of how many combinations of \(k\) items can be selected from \(n\) items without regard to order.
The formula for the binomial coefficient is:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]The binomial coefficient appears often in statistics, probability, and algebra, especially in the expansion of binomials, hence its name. It provides the critical information needed for calculating combinations in exam problems or any scenario where a selection process is involved.
For practical application, consider these elements:

• Understand the context of the problem to apply \( \binom{n}{k} \) correctly.
• Factorials in the formula help in the division and calculation part.
• The binomial coefficient is symmetrical, meaning \( \binom{n}{k} = \binom{n}{n-k} \).

By understanding and applying the binomial coefficient, you can simplify and solve complex combinatorial problems.