Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, compute the value of the expression. $$ C(7,6) $$

Step-by-Step Solution

Answer

The value of \( C(7, 6) \) is 7.

Step 1: Understand the Combination Notation

The expression \( C(n, k) \) represents the number of combinations of \( n \) items taken \( k \) at a time. This is calculated using the formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n! \) denotes the factorial of \( n \).

Step 2: Identify the Values of n and k

In the given expression \( C(7, 6) \), we identify \( n = 7 \) and \( k = 6 \).

Step 3: Substitute Values into the Combination Formula

Substitute the values of \( n \) and \( k \) into the formula: \[ C(7, 6) = \frac{7!}{6!(7-6)!} \]

Step 4: Calculate the Factorials

Calculate the factorials: \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \), \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \), and \( 1! = 1 \).

Step 5: Perform the Calculations

Plug the factorial values into the equation: \[ C(7, 6) = \frac{5040}{720 \times 1} = \frac{5040}{720} \] After performing the division, \( \frac{5040}{720} = 7 \).

Key Concepts

Factorials

Factorials are a fundamental concept in combinatorics and are used to calculate permutations and combinations. The notation for a factorial is an exclamation mark after a number, such as in \( n! \). Factorials represent the product of an integer and all the positive integers below it.
For instance, \( 5! \) is calculated as follows:

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

Factorials grow very rapidly, which is important to keep in mind when calculating combinations or permutations involving larger numbers. The factorial of zero is defined to be \(1\), which is useful in computation. Understanding factorials is crucial because they are a core component of the formulas for permutations and combinations, serving as the basis for these calculations.

Combination Formula

Combinations are a way to select items from a larger set where the order does not matter. The combination formula is used to find the total number of possible combinations of \( n \) items taken \( k \) at a time, denoted as \( C(n, k) \).
To compute this, we use the formula:

• \( C(n, k) = \frac{n!}{k!(n-k)!} \)

This formula involves factorials, as it divides the factorial of the total number of items by the factorial of the number of items chosen times the factorial of the remaining items. Using this formula helps in calculating how many different ways a subset of items can be selected, which is particularly useful in fields such as probability and statistics.
For example, \( C(7, 6) \) represents how many ways we can choose 6 items from a set of 7. Simplifying this expression, as shown in the example, illustrates the beauty of combinations: calculate the relevant factorials and simplify to find the result.

Permutations

Permutations are different from combinations because the order of the items does matter. In permutations, we determine how many ways we can arrange \( n \) items, which is represented by \( P(n, k) \), where \( n \) is the total number of items, and \( k \) is the number of items to arrange.
The formula for permutations is:

• \( P(n, k) = \frac{n!}{(n-k)!} \)

This formula highlights that a permutation is a rearrangement of items, unlike combinations where arrangement does not matter. As a result, permutations typically result in a larger number than combinations because they consider all possible sequences.
For example, with five books, the number of ways to arrange all of them in a sequence would be calculated as \( P(5, 5) = 5! \), which equals 120 different arrangements. This highlights the usefulness of permutations in scenarios where sequence or order is important, like seating arrangements or scheduling.