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Chapter 6: Problem 2

Evaluate the binomial coefficient. $$\left(\begin{array}{l} 8 \\ 6 \end{array}\right)$$

Short Answer

Expert verified
So, the value of the binomial coefficient \(\left(\begin{array}{l} 8 \ 6 \end{array}\right)\) is 28.

Step by step solution

01

Identify n and k

To begin we identify the values of n and k from the given binomial coefficient \(\left(\begin{array}{l} 8 \ 6 \end{array}\right)\). Here, n is 8 and k is 6.
02

Apply the formula

We then substitute the values of n and k into the formula for the binomial coefficient \( C(n, k) = \frac{n!}{k!(n-k)!} \). Doing so, we get \( C(8, 6) = \frac{8!}{6!(8-6)!} \).
03

Calculate the factorials

Next, we calculate the factorial of each number. A factorial is the product of all positive integers up to that number. So, \(8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320\), \(6! = 6 * 5 * 4 * 3 * 2 * 1 = 720\), and \(2! = 2 * 1 = 2.\)
04

Substitute the factorials into the formula

Substituting the factorial values into the formula from step 2 gives us: \(C(8, 6) = \frac{40320}{720 * 2} \).
05

Carry out the division

Finally, we divide, to find that \(C(8, 6) = \frac{40320}{1440} = 28.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial
A factorial is a fundamental concept in mathematics, denoted by an exclamation mark (!). It's crucial for understanding binomial coefficients and many other areas.
For a given positive integer, say 'n', the factorial, denoted as \( n! \), is the product of all positive integers from 1 to n.
Here's how it works:
  • \(0! = 1\) by definition.
  • \(1! = 1\)
  • For any n > 1, \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \)
For instance, to find \( 5! \), you would calculate: \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow incredibly quickly, which is why they are essential for calculations involving large numbers, such as in combinatorics and permutations.
Combinations
Combinations are a key concept in combinatorics, specifically when we are interested in selecting items from a group without considering the order of selection.
When faced with such a problem, we use the concept of a binomial coefficient to determine the number of combinations.
The formula for finding combinations, or the binomial coefficient, is \( C(n, k) = \frac{n!}{k!(n-k)!} \):
  • 'n' represents the total number of items.
  • 'k' is the number of items to be chosen.
  • The expression \( (n-k)! \) accounts for the items not chosen.
Using this formula, you can solve problems that ask for different ways to choose a subset of items from a larger set where order does not matter. This is fundamental in fields ranging from statistics to probability theory.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations, permutations, and the counting of various structures. It's a foundational pillar in mathematical problem-solving and helps us understand different possible groupings and orders in a set.
Within combinatorics:
  • Permutations involve arrangements where order is significant.
  • Combinations, on the other hand, disregard order, making them a simpler case of permutations.
By applying these principles, you can solve complex problems related to arrangements, selections, and optimizations. Binomial coefficients are a popular topic within combinatorics, offering a way to calculate combinations, as seen in our exercise. This branch is not just academic; it's used in computer science, algorithm design, and even in solving logistical problems.

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