Problem 23 - Precalculus: Mathematics for Calculus

Question

Evaluate the expression. $$\left(\begin{array}{l}5 \\ 0\end{array}\right)+\left(\begin{array}{l}5 \\\ 1\end{array}\right)+\left(\begin{array}{l}5 \\\ 2\end{array}\right)+\left(\begin{array}{l}5 \\\ 3\end{array}\right)+\left(\begin{array}{l}5 \\\ 4\end{array}\right)+\left(\begin{array}{l}5 \\ 5\end{array}\right)$$

Step-by-Step Solution

Answer

The value of the expression is 32.

Step 1: Understand the Expression

The expression consists of several combinations, each represented as \( \binom{n}{k} \), which is the binomial coefficient. In this case, each term is \( \binom{5}{k} \) for \( k = 0, 1, 2, 3, 4, 5 \).

Step 2: Apply the Binomial Coefficient Formula

The binomial coefficient is calculated using the formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). We will use this formula to compute each term in the expression.

Step 3: Calculate Each Term

1. Calculate \( \binom{5}{0} \): \( \binom{5}{0} = \frac{5!}{0!(5-0)!} = 1 \)2. Calculate \( \binom{5}{1} \): \( \binom{5}{1} = \frac{5!}{1!(5-1)!} = 5 \)3. Calculate \( \binom{5}{2} \): \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = 10 \)4. Calculate \( \binom{5}{3} \): \( \binom{5}{3} = \frac{5!}{3!(5-3)!} = 10 \)5. Calculate \( \binom{5}{4} \): \( \binom{5}{4} = \frac{5!}{4!(5-4)!} = 5 \)6. Calculate \( \binom{5}{5} \): \( \binom{5}{5} = \frac{5!}{5!(5-5)!} = 1 \)

Step 4: Sum the Calculated Terms

Now, add up all the results from the binomial coefficient calculations: \( 1 + 5 + 10 + 10 + 5 + 1 \).

Step 5: Compute the Final Sum

The sum of the terms is: \( 1 + 5 + 10 + 10 + 5 + 1 = 32 \).

Key Concepts

Binomial Coefficient

The binomial coefficient is a fundamental part of mathematics, primarily used to denote the number of ways to choose \( k \) elements from a set of \( n \) elements without considering the order. Often represented as \( \binom{n}{k} \), this concept is crucial in fields like combinatorics. For instance, in our exercise, we are considering all the ways to select 0 to 5 elements from a 5-element set. The formula for calculating a binomial coefficient is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]This equation uses factorials (explained in another section) to compute the desired value. What's handy about the binomial coefficient is its application in the binomial theorem, which aids in expanding expressions raised to a power, like \( (a+b)^n \). Remember, this coefficient only makes sense within non-negative integers where \( 0 \leq k \leq n \). This ensures that we are talking about valid counts of selections from a set.

Combinatorics

Combinatorics is the field of mathematics focused on counting, arrangement, and combination. It helps solve problems related to the selection and arrangement of objects, which might seem straightforward, but can become tricky with larger sets. Imagine you want to understand the different ways to organize a set of items. It's combinatorics that will give you tools like permutations and combinations to find the number of different possible arrangements or selections without repeating elements. Permutations refer to arranging all the elements of a set into some sequence or order. In contrast, combinations refer to selection without regard to the order. In our context, combinations are relevant because we aren't concerned about the order in which the elements are chosen, just how many ways we can choose them.

Factorial

Factorial, denoted by an exclamation mark (\(!\)), is a simple yet powerful concept in mathematics. The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \). For example, the factorial of 5, written as \( 5! \), is calculated as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]Factorial is essential in calculations involving permutations and combinations, providing a way to express the number of ways in which a set of elements can be arranged. For instance, in the binomial coefficient formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), factorials simplify complex multiplicative sequences into a single term, making lengthy calculations manageable.It's important to know that \( 0! = 1 \) by convention, which simplifies formulas and calculations that involve choosing 0 items from a set.