Problem 20 - Precalculus: Mathematics for Calculus

Question

Evaluate the expression. $$\left(\begin{array}{c}10 \\ 5\end{array}\right)$$

Step-by-Step Solution

Answer

252

Step 1: Understand the Problem

The expression \( \left( \begin{array}{c} 10 \ 5 \end{array} \right) \) is a binomial coefficient, also denoted as \( C(10, 5) \) or \( \binom{10}{5} \). It represents the number of ways to choose 5 items from a set of 10 without considering the order.

Step 2: Apply the Binomial Coefficient Formula

The formula for a binomial coefficient is given by:\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, \( n = 10 \) and \( r = 5 \). Substitute these values into the formula to evaluate \( \binom{10}{5} \).

Step 3: Calculate Factorials

To solve \( \binom{10}{5} \), calculate the factorials involved:- \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)- \( (10-5)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

Step 4: Substitute Factorials into the Formula

Substitute the factorials into the binomial coefficient formula:\[ \binom{10}{5} = \frac{10!}{5! \times 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \]

Step 5: Simplify the Expression

Now simplify the expression by canceling out terms:\[ \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \]- Start simplifying the numerator: - 10 divided by 5 equals 2. - Simplify: \( 2 \times 9 \times 8 \times 7 \times 6 \) divided by \( 4 \times 3 \times 2 \times 1 \). - Result: \( 3024 \div 120 = 252 \).

Step 6: Final Answer

The value of the binomial coefficient \( \left( \begin{array}{c} 10 \ 5 \end{array} \right) \) is 252. This means there are 252 ways to choose 5 items from 10.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, especially in combinatorics. A factorial, denoted by an exclamation mark (!), is the product of all whole numbers from a given number down to 1. For instance, 5 factorial, written as 5!, is calculated as:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
This concept is used frequently in various mathematical formulas, including permutations and combinations. Factorials grow very rapidly, meaning they become large numbers even for relatively small inputs. Factorials are integral in calculating probabilities and possible arrangements of objects.
When factorials are used in combination with division, as seen in the binomial coefficient, they help to eliminate repeating calculations, effectively reducing the complexity of counting algorithms. This makes operations like determining the number of combinations feasible, even for large numbers of elements.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns among objects. It is essential for understanding problems involving selections and arrangements. Often, we deal with problems such as "How many different ways can you arrange a set of objects?" or "How many ways can you choose a subset from a larger set?"
One of the fundamental principles in combinatorics is understanding how to count without direct enumeration, which involves using formulas to compute numbers efficiently. The use of factorials in combinatorics, especially in permutations (arrangements) and combinations (selections), is crucial. They allow us to perform calculations that would be impractical by hand.
A vital tool in combinatorics is the concept of the binomial coefficient, which gives us the number of ways to select a subset of items from a larger set, without regard to order. This helps solve problems related to arrangements, distributions, and selections across different mathematical disciplines, including probability theory.

Binomial Theorem

The Binomial Theorem is a significant principle in mathematics, especially in algebra and combinatorics, which provides a quick way to expand expressions of the form \((a + b)^n\). The theorem states that:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]
In this expression, \(\binom{n}{k}\) represents the binomial coefficient, which tells us how many ways we can select \(k\) elements from \(n\) elements. This coefficient plays a key role in the expansion of the binomial expression, as it determines the multiplicative factor of each term in the expansion.
The Binomial Theorem is particularly useful for calculating powers of binomial expressions without multiplying everything directly, which is especially helpful when dealing with large numbers or higher powers. It not only finds applications in theoretical mathematics but also in fields like computer science, physics, and statistics, where one often encounters probabilistic models involving binomial conditions.