Problem 20 - College Algebra

Question

For the following exercises, compute the value of the expression. $$ C(8,5) $$

Step-by-Step Solution

Answer

The value of the expression \( C(8,5) \) is 56.

Step 1: Understanding the Problem

The expression \( C(8,5) \) refers to the number of combinations of 8 items taken 5 at a time without regard to order. It is calculated using the combination formula.

Step 2: Applying the Combination Formula

The combination formula is \( C(n, r) = \frac{n!}{r!(n - r)!} \). For this problem, \( n = 8 \) and \( r = 5 \). Substitute these values into the formula: \( C(8,5) = \frac{8!}{5!(8 - 5)!} \).

Step 3: Calculating Factorials

Calculate the factorials needed: - \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) - \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \) - \( 3! = 3 \times 2 \times 1 \).

Step 4: Simplifying the Expression

Substitute the factorials into the formula: \( C(8,5) = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1 \times 3 \times 2 \times 1} \). Simplify by canceling out common terms in the numerator and denominator.

Step 5: Performing Division

After simplification, the expression reduces to \( \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \). Calculate it step by step: - \( 8 \times 7 = 56 \) - \( 56 \times 6 = 336 \) - \( 336 \div 6 = 56 \).

Step 6: Conclusion

The value of the expression \( C(8,5) \) is 56.

Key Concepts

Combination Formula

In combinatorics, the combination formula is a vital tool used to determine the number of ways to choose a subset of items from a larger pool. The order of selection does not matter in combinations, unlike permutations. The formula is expressed as:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]Here,

• \( n \) represents the total number of items available.
• \( r \) is the number of items to be chosen.

This formula calculates how many ways you can pick \( r \) items from \( n \) items. It is crucial in scenarios where the arrangement is not important—for example, selecting lottery numbers or forming groups in a classroom.To solve \( C(8,5) \), we substitute 8 for \( n \) and 5 for \( r \) into the formula, which helps us find how many different groups of 5 items can be chosen from a total of 8 items.

Factorials

Factorials appear prominently in the combination formula, understood as the product of all positive integers up to a given number \( n \). When written, a factorial is denoted by an exclamation mark \( n! \). For example:- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)- \( 3! = 3 \times 2 \times 1 = 6 \)Factorials grow rapidly with the number, and understanding how to compute them is essential for solving many problems in mathematics, particularly when dealing with combinations and permutations.In our problem, we calculate:

• 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320
• 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
• 3! = 3 \times 2 \times 1 = 6

Understanding the factorial function simplifies the process of calculating binomial coefficients.

Binomial Coefficient

The binomial coefficient, represented as \( C(n, r) \), is the result obtained from the combination formula and is typically denoted by the binomial symbol "\( \binom{n}{r} \)". It represents the number of combinations possible, which relates to probabilities and various counting problems in mathematics.To compute \( C(8,5) \) or \( \binom{8}{5} \), based on the work from the combination formula:We have \[ C(8, 5) = \frac{8!}{5! \times 3!} = \frac{40,320}{120 \times 6} \]We simplify the equation by canceling out the factorials as shown:- The numerator's 8! = 40,320- The denominator is the product of \( 5! \) and \( 3! \), which gives 720Dividing the two results:\[ \frac{40,320}{720} = 56 \]Thus, \( C(8,5) = 56 \), indicating there are 56 different ways to choose 5 items from a set of 8.