Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ \left(\begin{array}{l}{15} \\ {10}\end{array}\right) $$

Step-by-Step Solution

Answer

3003

Step 1: Understand the Notation

The expression \( \left(\begin{array}{c}{n} \ {r}\end{array}\right) \) is a binomial coefficient, also known as 'n choose r'. It represents the number of ways to choose \( r \) objects from \( n \) without regard to the order. The formula is \( \frac{n!}{r!(n-r)!} \).

Step 2: Identify the Values

In this problem, \( n = 15 \) and \( r = 10 \). We will substitute these values into the binomial coefficient formula.

Step 3: Substitute the Values into the Formula

Substitute \( n = 15 \) and \( r = 10 \) into the formula: \[ \left(\begin{array}{c}{15} \ {10}\end{array}\right) = \frac{15!}{10!(15-10)!} = \frac{15!}{10!5!} \]

Step 4: Calculate the Factorials

Compute the factorials needed: - \( 15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)

Step 5: Simplify the Expression

The expression becomes: \[ \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} \]Calculate each part separately and simplify the division.

Step 6: Perform the Calculations

Calculate: \[ 15 \times 14 \times 13 \times 12 \times 11 = 360360 \]And \[ 5 \times 4 \times 3 \times 2 \times 1 = 120 \]Thus, \[ \frac{360360}{120} = 3003 \]

Step 7: Conclude the Result

The result of the binomial coefficient \( \left(\begin{array}{c}{15} \ {10}\end{array}\right) \) is 3003.

Key Concepts

Factorial

In mathematics, a factorial is a function that multiplies a sequence of descending natural numbers down to 1. It is denoted by an exclamation mark, as in \( n! \) for a positive integer \( n \). This function is fundamental in various mathematical disciplines, such as algebra and combinatorics, due to its ability to calculate permutations and combinations.
For instance, the factorial of 5, written as \( 5! \), is calculated as follows:

• Start with 5.
• Multiply by 4 (the next lower number).
• Continue this process down to 1.

The operation looks like: \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). The factorial grows very quickly with larger numbers. That's why computer programs are often used for calculating larger factorials efficiently.
Factorials are crucial for defining permutations (arrangements) and combinations (selections), both of which are key elements in probability and combinatorial problems.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arranging, and analyzing discrete structures. It explores various methods to count combinations and permutations of specific sets, crucial for problem-solving scenarios in algebra, probability, and beyond.
A primary tool in combinatorics is the binomial coefficient, which helps determine the number of ways to choose a subset of items from a larger set. Mathematically, it is represented as \( \binom{n}{r} \), known as 'n choose r'. The formula \( \frac{n!}{r!(n-r)!} \) calculates how many combinations of \( r \) items can be selected from a set of \( n \) items.

• \( n! \) accounts for the total number of arrangements of a set.
• \( r! \) adjusts for the arrangements within the chosen subset.
• \( (n-r)! \) accounts for the remaining elements.

For example, in the original exercise, evaluating \( \binom{15}{10} \) involves determining how many ways 10 items can be chosen from 15. This specific binomial coefficient can solve real-world problems like seating arrangements or team selections, demonstrating the practical applications of combinatorics.

Algebra 2

Algebra 2, a fundamental mathematics course in many high school curricula, delves into complex equations and functions, including polynomial equations, exponential functions, and logarithms. This course forms the basis for advanced studies in mathematics and other fields like physics and engineering.
Within Algebra 2, students are introduced to concepts such as sequences and series, where factorials and binomial coefficients frequently appear. Understanding these ideas is crucial for solving complex algebraic equations and analyzing patterns.
In the context of Algebra 2, students might encounter problems involving real-life contexts, such as calculating probabilities or determining possible combinations and arrangements. For example, when working with the binomial theorem, which expands powers of binomials, students will often use binomial coefficients. This highlights the interconnection between algebraic concepts and combinatorial methods.
By grasping these fundamentals, students can develop critical problem-solving skills and a deeper understanding of how mathematical concepts apply beyond the classroom, ensuring they are well-prepared for more advanced mathematical studies.