Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\left(\begin{array}{c}10 \\\8\end{array}\right)$$

Step-by-Step Solution

Answer

The value of \( \binom{10}{8} \) is 45.

Step 1: Understanding the Binomial Coefficient

The expression \( \binom{10}{8} \) is a binomial coefficient, which represents the number of ways to choose 8 elements from a set of 10 elements without regard to order.

Step 2: Formula for Binomial Coefficient

The formula for the binomial coefficient \( \binom{n}{k} \) is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] where \( n! \) represents the factorial of \( n \).

Step 3: Substitute Values into the Formula

Substitute \( n = 10 \) and \( k = 8 \) into the formula:\[\binom{10}{8} = \frac{10!}{8! \, (10-8)!}\] which simplifies to \( \frac{10!}{8! \, 2!} \).

Step 4: Calculate the Factorials

Compute the factorial values: \( 10! = 10 \times 9 \times 8! \) and \( 2! = 2 \times 1 = 2 \).

Step 5: Simplifying the Expression

The equation becomes:\[\frac{10 \times 9 \times 8!}{8! \, 2}\]The \( 8! \) terms cancel out, leaving:\[\frac{10 \times 9}{2}\]

Step 6: Final Calculation

Simplify the remaining expression:\[\frac{10 \times 9}{2} = \frac{90}{2} = 45\] So, \( \binom{10}{8} = 45 \).

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics focused on counting, arranging, and analyzing the structure of sets. It's like a mathematical toolkit designed to solve problems where you need to find out how many ways you can organize or choose items under specific constraints. This might sound simple, but when dealing with large sets or complex conditions, it becomes quite challenging and exciting!

In problems involving combinatorics, you'll often come across binomial coefficients, which are helpful in counting how many combinations are possible without worrying about order. Think of it as figuring out how to split a deck of cards or arranging books on a shelf in distinct ways. Combinatorics isn't just useful in pure mathematics, but also in fields like computer science, probability, and even in day-to-day decision-making.

Factorial

The concept of factorial, denoted by the symbol "!", plays a crucial role in combinatorics, especially when calculating permutations and combinations. A factorial of a non-negative integer is the product of all positive integers less than or equal to that number. It provides a way to determine the number of ways to arrange a set of objects.

Here's a simple breakdown:

• The factorial of 1, written as 1!, equals 1.
• The factorial of 2, written as 2!, is 2 (or 2 x 1).
• Generally, for any positive integer n, we write n! = n x (n - 1) x (n - 2) x ... x 1.

Factorials grow very fast. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. When evaluating expressions like binomial coefficients, factorials help to systematize the counting process, especially by canceling out common terms to simplify calculations.

Permutation and Combination

Permutations and combinations are fundamental principles of counting in combinatorics. Both involve selecting items from a group but differ in whether the order of the selection matters or not.

Permutations are used when the order does matter. Imagine a scenario where you are determining possible arrangements for a set of books on a shelf. Each different order counts as a unique permutation.

Combinations, however, come into play when order does not matter. They are used to select items where the sequence is irrelevant, like choosing a team from a group where all selections can be deemed identical in importance, regardless of which order they were picked in.

In mathematics, combinations are calculated using binomial coefficients, a reflection of selecting subgroups in an unordered manner. The formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) clearly showcases this, guiding how many different ways you can select \( k \) items from \( n \) without caring about arrangement.