Problem 51 - Algebra and Trigonometry

Question

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

There are 120 ways to choose 7 questions from 10.

Step 1: Identify the Combination Formula

To solve the problem of choosing 7 questions from 10, we use the combination formula. It is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Here, \( n = 10 \) and \( r = 7 \).

Step 2: Plug Values into the Formula

Substitute \( n = 10 \) and \( r = 7 \) into the combination formula: \( C(10, 7) = \frac{10!}{7!(10-7)!} \).

Step 3: Simplify the Denominator

Calculate the denominator: \( 7!(10-7)! = 7! \, 3! \). We know \( 7! = 5040 \) and \( 3! = 6 \). Multiply them: \( 7! \, 3! = 5040 \, \times 6 = 30240 \).

Step 4: Simplify the Numerator

Calculate the numerator \( 10! \). To simplify: \( 10! = 10 \times 9 \times 8 \times 7! \).

Step 5: Cancel Out Common Factorials

Both in the numerator and the denominator, \( 7! \) appears. They can be canceled out, resulting in \( \frac{10 \times 9 \times 8}{3!} \).

Step 6: Calculate the Result

Calculate \( \frac{10 \times 9 \times 8}{6} \). Perform the multiplication and division step-by-step: \( 10 \times 9 = 90 \), then \( 90 \times 8 = 720 \), and finally \( \frac{720}{6} = 120 \).

Step 7: Conclusion

Therefore, the number of ways the student can choose 7 out of the 10 questions is 120.

Key Concepts

Combination Formula

The combination formula is a mathematical way to calculate how many different groups, or combinations, can be formed by choosing a smaller number of items from a larger group, without considering the order of selection. This is essential when you have a set and need to determine how many ways there are to pick a subset from it.

The formula is written as:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

where:

• \( n \) is the total number of items to choose from.
• \( r \) is the number of items to select.
• The expression \( n! \) represents the factorial of \( n \).

Let's break it down simply. You first calculate \( n! \), which is the factorial of the total number of items. Then, you divide it by the product of \( r! \) and \((n-r)!\). This helps you count the different ways to select \( r \) items, ensuring no repetition and order doesn't matter.

Factorial Calculation

Factorials are a fundamental part of the combination formula, and understanding them makes solving combination problems more accessible. A factorial, noted as \( n! \), is the product of all positive integers up to \( n \). So, for a number \( n \), the factorial is obtained by multiplying all integers from 1 up to \( n \).

For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Factorials grow very fast, and they are mainly used in mathematics to simplify calculations in combinatorics, like permutations and combinations.

In the context of our exercise, factorials are used to calculate the total number of ways certain items can be chosen from a set, as we saw with \( 10! \), \( 7! \), and \( 3! \).

Permutations and Combinations

Permutations and combinations are concepts in mathematics used to count the different ways to arrange or select items.

• Permutations: All about arrangement. When order matters, it's a permutation. For example, arranging books on a shelf.
• Combinations: All about selection. When order doesn't matter, it's a combination. Like choosing friends to invite to a party.

In our exercise, since the order of the exam questions does not matter, we are dealing with a combination.
The distinction is crucial; permutations consider the arrangement as important, leading to a larger number of outcomes compared to combinations where only selection matters. Here, combinations allow us to count how many ways we can choose without the sequence being important.

Binomial Coefficient

The binomial coefficient is a central part of combinatorics and is denoted as \( \binom{n}{r} \), equivalent to \( C(n, r) \) in the combinations formula. It signifies the number of ways to choose \( r \) items from \( n \) items without regard to order, exactly what we are calculating in our exercise.

The binomial coefficient can be calculated using the combination formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

In the context of probabilities, it also appears in the expansion of binomial expressions, which is why it's called a binomial coefficient. Thus, understanding this concept is not only useful for counting combinations but also in other advanced mathematical contexts.

In our example, calculating \( \binom{10}{7} \) gives us the actual number of ways to choose 7 questions from a total of 10, which was calculated to be 120. This represents the binomial coefficient of 10 choose 7.