Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the binomial coefficient. \(\left(\begin{array}{c}10 \\ 4\end{array}\right)\)

Step-by-Step Solution

Answer

The binomial coefficient \(\left(\begin{array}{c}10 \ 4\end{array}\right)\) is 210.

Step 1: Identify values of n and k

First, identify the values of n and k from the given binomial coefficient. Here, n = 10 and k = 4

Step 2: Substitute into formula

Next, substitute n and k into the formula for binomial coefficient. That gives you \( \left(\begin{array}{c}10 \ 4\end{array}\right) = \frac{10!}{4!(10-4)!} \)

Step 3: Calculate factorial values

Now, calculate the factorial values. \(10! = 3,628,800\), \(4! = 24\), and \((10-4)! = 6!. 6! = 720 \)

Step 4: Substitute factorial values and simplify

Substitute the factorial values back into the formula and simplify: \(\frac{3,628,800}{24*720} = 210 \)

Key Concepts

Factorials in Binomial Coefficients

The binomial coefficient is a fundamental concept in combinatorics that counts the number of ways to choose a subset of k elements, disregarding the order, from a set of n elements. The notation \(\left(\begin{array}{c}n \ k\end{array}\right)\) represents the binomial coefficient, also known as 'n choose k'.

The calculation of a binomial coefficient involves factorials, which are denoted by an exclamation point and represent the product of all positive integers up to a given number. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

In binomial coefficients, factorials are used to ensure that the order of selection does not matter. The general formula for a binomial coefficient is given by \(\left(\begin{array}{c}n \ k\end{array}\right) = \frac{n!}{k! (n-k)!}\), where \(n!\) is the factorial of the total number of elements, \(k!\) is the factorial of the number of elements chosen, and \(n-k\)! is the factorial of the difference between the total number of elements and the number chosen.

When calculating binomial coefficients, understanding how to compute factorials is crucial. It's important to observe that values of factorials grow very quickly with increasing numbers, so simplifying them correctly within binomial coefficients is key to finding the correct combination values efficiently.

Combinatorics

Combinatorics is an area of mathematics primarily concerned with counting, both as a means for describing complex structures and to calculate probabilities. It includes various topics, such as combinations, permutations, and binomial coefficients.

In combinatorics, the concept of combinations is used when the order of selection does not matter. This is where binomial coefficients come into play as they express the number of combinations that can be made by selecting k elements from a set of n distinct elements.

Understanding combinatorics enhances problem-solving abilities in various fields, from computer science to biology. It's not only about how many ways something can occur but also about constructing and analyzing complex structures, optimizing arrangements, and solving logical puzzles.

Tools from combinatorics are used to solve problems that involve finding probabilities, minimum requirements, or maximal capacity, and they can get quite intricate as the size of the sets increases. Clear comprehension of factorial fundamentals, the principles of binomial coefficients, and Pascal's Triangle are instrumental in mastering combinatorial techniques.

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers that gives the coefficients of the expansion of a binomial expression. It's a visual representation of binomial coefficients arranged in rows and columns, where each number is the sum of the two directly above it. The numbers on the outermost diagonals are always 1, and any other number can be found by adding the two numbers from the row above that are positioned to the immediate left and right.

To visualize, the rows of Pascal's Triangle start with 1 at the top. Each subsequent row begins and ends with 1, and any entry in the triangle is the sum of the two entries from the row above:      1    1  1  1  2  1 1  3  3  11  4  6  4  1Each row corresponds to the powers of 11 (for the first few rows), and the numbers directly give the binomial coefficients for the expansion \( (a + b)^n \). Pascal’s Triangle is a simple, yet powerful tool that demonstrates several mathematical concepts, including binomial expansions, combinatorial numbers, and properties concerning probability theory.

It can be used as a quick reference to find binomial coefficients without resorting to the factorial formula, which is particularly useful with larger numbers where factorial calculations become computationally intensive.