Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Evaluate the expression. $$ \left(\begin{array}{c} 13 \\ 4 \end{array}\right) $$

Step-by-Step Solution

Answer

The value of \( \left(\begin{array}{c} 13 \\ 4 \end{array}\right) \) is 715.

Step 1: Understand the Notation

The expression \( \left(\begin{array}{c} 13 \ 4 \end{array}\right) \) is a binomial coefficient, also known as "13 choose 4", and represents the number of ways to choose 4 elements from a set of 13 elements.

Step 2: Use the Binomial Coefficient Formula

The binomial coefficient is given by the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). In our case, \( n = 13 \) and \( k = 4 \). Thus, \( \binom{13}{4} = \frac{13!}{4!(13-4)!} = \frac{13!}{4! \times 9!} \).

Step 3: Calculate the Factorials

We expand the factorials: \( 13! = 13 \times 12 \times 11 \times 10 \times 9! \). Since \( 9! \) appears in both the numerator and denominator, they cancel out. Thus the expression simplifies to: \( \frac{13 \times 12 \times 11 \times 10}{4!} \).

Step 4: Calculate \(4!\)

Calculate \( 4! \) (the factorial of 4): \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Step 5: Simplify the Expression

Now simplify \( \frac{13 \times 12 \times 11 \times 10}{24} \). First compute the product \( 13 \times 12 \times 11 \times 10 = 17160 \). Then divide by 24: \( \frac{17160}{24} = 715 \).

Step 6: Conclusion

The value of \( \left(\begin{array}{c} 13 \ 4 \end{array}\right) \) is 715.

Key Concepts

Factorials

Factorials are a key mathematical concept often utilized in permutations and combinations. It is essential to understand that a factorial of a non-negative integer, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 13! = 13 \times 12 \times 11 \times 10 \times \cdots \times 2 \times 1 \)

Factorials grow very quickly, which is why they are so useful in counting methods in mathematics. In calculations involving binomial coefficients, like in the exercise, it's important to cancel out as many common factorial terms as possible to simplify the problem and avoid dealing with very large numbers.

Combinatorics

Combinatorics is the field of mathematics concerned with counting, arranging, and selecting objects. It is a foundational concept when discussing binomial coefficients. One of the core ideas in combinatorics is that of a combination. This involves selecting items from a larger set where the order does not matter.
In the exercise given, we use a binomial coefficient to determine the number of combinations possible for choosing 4 elements from a set of 13. This is expressed as \( \binom{13}{4} \). Combinatorics essentially is about understanding different strategies to count objects efficiently and accurately, leading to solutions for such problems effectively.

Binomial Theorem

The binomial theorem is a powerful mathematical tool used to expand expressions raised to a power. The theorem states:

• \( (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \)

This relates directly to binomial coefficients, which are the building blocks of the expansions. These coefficients tell us how many ways we can choose \( k \) items from \( n \) choices, which fit directly into each term of the expansion.
In our specific exercise, although we are solely calculating \( \binom{13}{4} \) without further application, understanding the theorem helps contextualize why such calculations are significant in mathematics, especially in polynomial expansions. Hence, recognizing the relevance of each layer of the calculation is important in most algebraic and combinatorial contexts.