Problem 18 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \left(\begin{array}{l}{8} \\ {3}\end{array}\right) $$

Step-by-Step Solution

Answer

The expression evaluates to 56.

Step 1: Understand the Notation

The expression \( \left(\begin{array}{c}{8} \ {3}\end{array}\right) \) is a combination notation. It stands for "8 choose 3," which is the number of ways to choose 3 elements from a set of 8 elements without regard to order.

Step 2: Use the Combination Formula

The formula for a combination is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here, \( n = 8 \) and \( r = 3 \). Substitute the values into the formula: \( \binom{8}{3} = \frac{8!}{3! \times (8-3)!} \).

Step 3: Simplify the Factorials

Calculate the factorials in the denominator and numerator. \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \), but for simplification, we only need \( 8 \times 7 \times 6 \) because it will cancel out with \( 5! \) which is common in both the numerator and denominator: \( 8 \times 7 \times 6 \times 5! \). The denominator is \( 3! = 3 \times 2 \times 1 \) and \( 5! \). So, it becomes \( \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} \).

Step 4: Cancel Out the Common Terms

The \( 5! \) in the numerator and denominator cancels out, so we are left with \( \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \).

Step 5: Continue Simplification

Calculate the remaining expression: \( 8 \times 7 \times 6 = 336 \) and \( 3 \times 2 \times 1 = 6 \). Divide \( 336 \) by \( 6 \) to get the final result: \( \frac{336}{6} = 56 \).

Step 6: Final Result

The number of ways to choose 3 elements from a set of 8 is 56.

Key Concepts

Combination Formula

In combinatorics, we often want to know how many ways we can choose a subset of items from a larger set, without caring about the order. This is where the combination formula comes into play. The combination formula is mathematically represented as:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where:

• \( n \) is the total number of items,
• \( r \) is the number of items to choose,
• \( ! \) denotes the factorial of a number.

This formula helps us calculate the number of ways to choose \( r \) elements from a set of \( n \) elements. The order in which you choose does not matter. It is fundamental in problems where the setup involves selecting a group or committee from a larger population.

Factorials

Factorials are a key concept in permutations and combinations. The factorial of a number \( n \), denoted \( n! \), is the product of all positive integers from 1 to \( n \). For example, the factorial of 4 is:\[4! = 4 \times 3 \times 2 \times 1 = 24\]Factorials grow rapidly with larger numbers and are particularly useful in combinatorial mathematics. When applying the combination formula, factorials calculate the total number of ordered arrangements for any chosen elements. In our specific exercise, calculating smaller factorials helps simplify operations by canceling out terms between numerators and denominators, which streamlines finding combinations efficiently. Remember:

• Factorials only include positive whole numbers,
• \( 0! = 1 \) by definition, an essential concept in combinatorics.

Binomial Coefficient

The binomial coefficient represents the number of combinations possible in a set. It is expressed as \( \binom{n}{r} \), read as "n choose r." The formula for the binomial coefficient is directly tied to the combination formula:\[\binom{n}{r} = \frac{n!}{r! \times (n-r)!}\]Using binomial coefficients, we can easily solve combinations problems like the one in our exercise. For instance, if we need to determine how many ways we can pick 3 items from 8 different items, we calculate \( \binom{8}{3} \) which acts as the main vehicle to determine those possibilities.Binomial coefficients feature prominently in Pascal's Triangle and the binomial theorem. They help expand expressions and determine probabilities in binomial distributions, making them versatile in various mathematical and statistical applications.