Problem 8 - College Algebra with Modeling and Visualization

Question

Evaluate the expression. $$ \left(\begin{array}{l} 5 \\ 2 \end{array}\right) $$

Step-by-Step Solution

Answer

The expression evaluates to 10.

Step 1: Understanding the Problem

The expression is the binomial coefficient, commonly denoted as \( \binom{5}{2} \). This represents the number of ways to choose 2 elements from a set of 5 elements, and is solved using the combination formula.

Step 2: Apply the Combination Formula

The binomial coefficient formula is given by: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]For this problem, \( n = 5 \) and \( r = 2 \). Substitute these values into the formula to get: \[ \binom{5}{2} = \frac{5!}{2!(5-2)!} \]

Step 3: Calculate Factorials

Calculate the factorials needed:- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)- \( 2! = 2 \times 1 = 2 \)- \( (5-2)! = 3! = 3 \times 2 \times 1 = 6 \)

Step 4: Substitute and Simplify

Substitute the factorials back into the formula:\[ \binom{5}{2} = \frac{120}{2 \times 6} = \frac{120}{12} \]Simplify the fraction:\[ \frac{120}{12} = 10 \]

Step 5: Conclusion

The value of the combination \( \binom{5}{2} \) is 10.

Key Concepts

Binomial Coefficient

The binomial coefficient is a central concept in combinatorics. It represents the number of different ways to choose a subset of items from a larger set. The notation \( \binom{n}{r} \) is commonly used to represent the binomial coefficient, which tells us how many ways we can choose \( r \) elements from \( n \) distinct elements.

It's a key idea because it allows us to count combinations of objects without having to list them all individually. For example, \( \binom{5}{2} \) represents how many ways you can pick 2 items from a set of 5. This concept is essential for various fields like probability, statistics, and even computer science, where understanding combinations helps solve more complex problems.

• Binomial coefficients are always integers, as they count discrete entities.
• The value of \( \binom{n}{n} \) or \( \binom{n}{0} \) is always 1, because there's exactly one way to select all or none of the items.

Factorial

Factorials play a vital role when calculating binomial coefficients. The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers up to \( n \). For instance, \( 5! \) equals \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials grow rapidly, which is why they're handy for counting permutations and combinations. They're crucial in obtaining the denominator of the binomial coefficient formula. Factorials also appear in various mathematical fields, particularly in calculus and algebra, due to their fundamental properties.

• By definition, \( 0! \) is 1, providing a foundation to simplify equations.
• Factorials are used in probability to calculate permutations - arrangements of objects, where order matters.

Combination Formula

The combination formula is a tool used to calculate how many different ways you can choose a subset of objects, regardless of order. The formula is expressed as: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] This formula considers the removal of order from selections, making it different from permutations which include arrangements where order matters.

The combination formula simplifies complex counting problems by breaking down into components of factorials, enabling you to accurately determine possible selections with straightforward calculations. It's used extensively in probability, enabling predictions about events without exhaustive listing of outcomes.

• Utilizes three factorials: the total number of elements, the desired number of combinations, and the difference.
• Helps in evaluating real-world problems where selection is key - like team formations, probability predictions.