Problem 16 - Beginning and Intermediate Algebra

Question

Evaluate each binomial coefficient. $$\left(\begin{array}{l}4 \\\2\end{array}\right)$$

Step-by-Step Solution

Answer

The binomial coefficient \(\binom{4}{2}\) can be calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), which results in \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{1 \times 2} = 6\). Therefore, \(\binom{4}{2} = 6\).

Step 1: Understand the binomial coefficient formula

The formula to calculate a binomial coefficient is the following: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(n\) is the total number of elements in the set and \(k\) is the number of elements we want to choose. In this case, we have \(n = 4\) and \(k = 2\).

Step 2: Plug in the values into the formula

Now we need to plug in the given values into the formula: \[ \binom{4}{2} = \frac{4!}{2!(4-2)!} \]

Step 3: Compute the factorials

Next, calculate the factorials involved in the expression: \[ = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1)((4-2) \times 1)} \]

Step 4: Simplify the expression

Now, simplify the expression by canceling out common terms: \[ = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1)(2 \times 1)} = \frac{4 \times 3}{1 \times 2} \]

Step 5: Calculate the final result

Finally, calculate the resulting value: \[ \binom{4}{2} = \frac{4 \times 3}{1 \times 2} = 6 \] The binomial coefficient \(\binom{4}{2}\) is equal to 6.

Key Concepts

Factorials

Factorials play a vital role in calculating binomial coefficients. A factorial is denoted by an exclamation mark (!), and it represents the product of all positive integers up to a specific number. For example, the factorial of 4, written as 4!, is equal to \(4 \times 3 \times 2 \times 1 = 24\). It's important to note that the factorial of zero is defined as 1. This is essential when working with combinations and permutations.
Understanding how to compute each factorial component is crucial. In the binomial coefficient \(\binom{n}{k}\), you have \(n!\), \(k!\), and \((n-k)!\). You must carefully execute each factorial calculation because they form the building blocks of the division used to solve for the coefficient.
It's helpful to practice calculating factorials separately to become more comfortable with the process. This proficiency will make tackling more complex problems with factorials much smoother.

Combinatorics

Combinatorics is the branch of mathematics concerned with counting, both in finite and infinite cases. It is the foundation for solving problems that involve grouping and arranging objects. The binomial coefficient is a quintessential element in combinatorics, helping you determine the number of ways to choose \(k\) items from \(n\) items without regard to order.
To understand the concept better, think of combinations as ways to form a specific group. The formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) is used to calculate these combinations. The denominator \(k!(n-k)!\) ensures that arrangements of the same items are not over-counted, thus providing an accurate count of unique groupings.

• Use combinatorics to approach problems involving group selections.
• Remember that the order does not matter in combinations, unlike permutations where order is important.

Combinatorics is a key area in probability, statistics, and many fields that rely on efficient problem-solving using mathematical techniques.

Algebra

Algebra serves as the language of generalization in mathematics. It allows you to work with abstract concepts and solve equations through symbolic representation. The process of finding and simplifying the binomial coefficient occurred within an algebraic framework.
Using the factorial concept, we express the binomial coefficient in an algebraic formula. Each step involves algebraic manipulation, such as simplifying expressions and canceling terms in the numerator and denominator. For example, simplifying \(\frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1}\) is an algebraic operation involving cancelation and reduction of fractions.

• Practice these algebraic techniques to master simplifications.
• Understanding symbolic representations is crucial for handling abstract mathematical concepts.

Algebraic proficiency provides you with the tools to tackle various mathematical challenges beyond simple arithmetic calculations.