Problem 17 - Precalculus: Mathematics for Calculus

Question

Evaluate the expression. $$\left(\begin{array}{l}6 \\ 4\end{array}\right)$$

Step-by-Step Solution

Answer

The value is 15.

Step 1: Understand the Expression

This expression is a binomial coefficient, also known as "n choose k," which is represented as \( \binom{n}{k} \). It calculates the number of ways to choose \( k \) items from \( n \) items without considering the order. In this problem, \( n = 6 \) and \( k = 4 \).

Step 2: Use the Binomial Coefficient Formula

The formula for a binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Here, we will substitute \( n = 6 \) and \( k = 4 \) into the formula: \( \binom{6}{4} = \frac{6!}{4!(6-4)!} \).

Step 3: Calculate Factorials

We need to calculate \( 6! \), \( 4! \), and \( 2! \): - \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \) - \( 4! = 4 \times 3 \times 2 \times 1 = 24 \) - \( 2! = 2 \times 1 = 2 \)

Step 4: Substitute and Simplify

Substituting the factorials into the formula gives: \( \binom{6}{4} = \frac{720}{24 \times 2} = \frac{720}{48} \). Now, simplify \( \frac{720}{48} = 15 \).

Step 5: Conclude the Calculation

Thus, the value of the expression \( \left(\begin{array}{l}6 \ 4\end{array}\right) \) or \( \binom{6}{4} \) is 15.

Key Concepts

Factorials

Factorials are fundamental in mathematics, especially when dealing with permutations and combinations. The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). It's denoted by \( n! \).

For example, if you have \( 3! \), this means you calculate \( 3 \times 2 \times 1 = 6 \). In this process, you're multiplying a sequence of descending natural numbers. Factorials grow very quickly with bigger \( n \), meaning they become large numbers fast.

Factorials are crucial in the calculation of binomial coefficients, which we'll explore below. Knowing how to compute them fluently is an essential skill for solving problems in combinatorics and probability.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and grouping objects. It's widely used in fields that require discrete structures, such as computer science and statistical analysis. One of the primary concepts in combinatorics is finding out how many ways you can arrange or select things from a set.

In our context, combinatorics helps us determine how many different ways we can choose a subset of items from a larger set. For instance, if you're choosing two books from a shelf of five, combinatorics tells you how to find all possible pairs. Combinatorics encompasses tools like the principle of inclusion-exclusion, Pascal's triangle, and our focus here—the binomial coefficient, often expressed as "n choose k."

Engaging with these tools allows seamless solutions to complex real-world problems, from scheduling tasks to designing networks.

n choose k

'n choose k' is a shorthand expression used in combinatorial mathematics to denote the number of ways to choose \( k \) items from a set of \( n \) items, regardless of the order of selection. This is captured using the binomial coefficient \( \binom{n}{k} \).

The formula for calculating 'n choose k' is given by:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

This formula uses factorials to account for the number of possible selections and arrangements.

For instance, in evaluating \( \binom{6}{4} \), it involves finding how many different groups of 4 can be formed from 6 items. We apply the values of the factorials calculated as shown in the exercise: \( 6! \) for the total items, \( 4! \) for the chosen items, and \( (6-4)! \) for the unchosen items.

This concept not only applies to simple selection problems but is the foundation for more advanced topics like binomial expansions and probability distributions.