Problem 15 - Algebra and Trigonometry

Question

13–18 Evaluate the expression. $$C(11,4)$$

Step-by-Step Solution

Answer

The expression \( C(11,4) \) evaluates to 330.

Step 1: Understanding Combinations

The expression \( C(n, k) \) is used to represent the number of ways to choose \( k \) items from \( n \) items without regard to order. It is calculated using the formula \( C(n, k) = \frac{n!}{k!(n-k)!} \). For this exercise, \( n = 11 \) and \( k = 4 \).

Step 2: Apply the Formula

Plug in the values into the combination formula: \( C(11, 4) = \frac{11!}{4!(11-4)!} = \frac{11!}{4!7!} \).

Step 3: Evaluate 11 factorial

Calculate \( 11! = 11 \times 10 \times 9 \times 8 \times 7! \). Notice that \( 7! \) will cancel out in the final step, simplifying the calculation.

Step 4: Simplify the Factorials

Since \( 11! = 11 \times 10 \times 9 \times 8 \times 7! \), when divided by \( 7! \), the result is \( 11 \times 10 \times 9 \times 8 \). Now divide this by \( 4! \), where \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Step 5: Calculate the Result

Simplify \( \frac{11 \times 10 \times 9 \times 8}{24} \). Calculating step-by-step: \( 11 \times 10 = 110 \), \( 110 \times 9 = 990 \), \( 990 \times 8 = 7920 \), then \( \frac{7920}{24} = 330 \).

Key Concepts

Factorials

A factorial is a fundamental mathematical concept used primarily in permutations and combinations. Represented by an exclamation mark (!), the factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). For example:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

Factorials are used to calculate the number of ways to arrange a set of items. This is crucial in understanding permutations and combinations because these operations rely on factorials for computation.
Factorials grow very quickly with increasing \( n \). For instance, while \( 5! \) is relatively small, \( 10! \) already jumps to \( 3,628,800 \). When solving problems involving combinations, factorials help simplify and reduce large products by cancelling out terms, saving both time and calculation complexity.

Permutations and Combinations

Permutations and combinations are two essential concepts in probability and statistics that involve the arrangement of items. Permutation relates to arrangements where order matters. To calculate permutations, you use the factorial of the number of items you are arranging. For example, if you have 3 books and you want to know how many ways you can arrange them on a shelf, you calculate \( 3! \), which equals 6.
Meanwhile, combinations consider arrangements where order does not matter. This is where the binomial coefficient \( C(n, k) \) comes into play, which is calculated using the formula:\[C(n, k) = \frac{n!}{k!(n-k)!}\]In other words:

• \( n \) represents the total number of items.
• \( k \) represents the number of items to choose.

Combinations are used when order is irrelevant, such as selecting a team of 4 from 11 candidates where the arrangement within the team doesn’t matter.

Binomial Coefficients

Binomial coefficients arise in the context of binomial expansion and in problems involving combinations. The notation \( C(n, k) \) is synonymous with the binomial coefficient and is read as "\( n \) choose \( k \)."This coefficient is a fundamental part of the binomial theorem, which is used to expand expressions of the form \( (a + b)^n \). Each term in the expansion has a coefficient that is a binomial coefficient.To compute a binomial coefficient, you use the formula:\[C(n, k) = \frac{n!}{k!(n-k)!}\]This formula uses factorials to simplify the calculation.
Binomial coefficients are often found in Pascal's Triangle, where each number is the sum of the two numbers directly above it in the previous row. This visually represents how combinations form the coefficients within the triangle.