Problem 24 - 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 0.

Step 1: Understanding Binomial Coefficients

The expression involves calculating several binomial coefficients. A binomial coefficient \( \left( \begin{array}{c} n \ r \end{array} \right) \) can be calculated using the formula \( \frac{n!}{r!(n-r)!} \), where \( n! \) denotes factorial of \( n \).

Step 2: Calculate \( \left( \begin{array}{c} 5 \\ 0 \end{array} \right) \)

The calculation of \( \left( \begin{array}{c} 5 \ 0 \end{array} \right) \) is straightforward because any number choose 0 is 1, so \( \left( \begin{array}{c} 5 \ 0 \end{array} \right) = 1 \).

Step 3: Calculate \( \left( \begin{array}{c} 5 \\ 1 \end{array} \right) \)

Using the formula, we get: \( \left( \begin{array}{c} 5 \ 1 \end{array} \right) = \frac{5!}{1!(5-1)!} = \frac{5 \times 4!}{1 \times 4!} = 5 \).

Step 4: Calculate \( \left( \begin{array}{c} 5 \\ 2 \end{array} \right) \)

This is calculated as \( \left( \begin{array}{c} 5 \ 2 \end{array} \right) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{20}{2} = 10 \).

Step 5: Calculate \( \left( \begin{array}{c} 5 \\ 3 \end{array} \right) \)

Using the symmetric property of combination \( \left( \begin{array}{c} 5 \ 3 \end{array} \right) = \left( \begin{array}{c} 5 \ 2 \end{array} \right) = 10 \), because choosing 3 is the same as not choosing 2.

Step 6: Calculate \( \left( \begin{array}{c} 5 \\ 4 \end{array} \right) \)

Symmetrically, \( \left( \begin{array}{c} 5 \ 4 \end{array} \right) = \left( \begin{array}{c} 5 \ 1 \end{array} \right) = 5 \).

Step 7: Calculate \( \left( \begin{array}{c} 5 \\ 5 \end{array} \right) \)

Similarly, any number choose itself is 1, so \( \left( \begin{array}{c} 5 \ 5 \end{array} \right) = 1 \).

Step 8: Substitute and Evaluate the Expression

Substitute all calculated values: \( 1 - 5 + 10 - 10 + 5 - 1 \). Simplifying this, we proceed as \( (1 - 5 = -4), (-4 + 10 = 6), (6 - 10 = -4), (-4 + 5 = 1), (1 - 1 = 0) \).

Key Concepts

Factorials

A factorial, denoted by an exclamation point (!), is a way to multiply a series of descending natural numbers. For example, the factorial of 5, written as \(5!\), is calculated as:

• \(5! = 5 \times 4 \times 3 \times 2 \times 1\)
• This equals 120.

Factorials are a fundamental part of mathematics, particularly in combinatorics and permutations. They help in determining the number of ways to arrange a set of objects.
In combinatorics, factorials are crucial when calculating binomial coefficients, which are used to determine how many ways an "r" number of outcomes can be chosen from a set of "n" objects. For instance, the expression used in binomial coefficients is \( \frac{n!}{r!(n-r)!} \).
Factorials grow very quickly as the number increases. For example, \(10!\) is 3,628,800. This rapid growth means factorials are incredibly useful in real-world applications involving calculations with large numbers.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of objects. It encompasses many concepts, such as permutations, combinations, and binomial coefficients. In our exercise, we specifically deal with combinations.
A combination is the selection of items from a larger set such that the order of selection doesn't matter. For binomial coefficient \( \left( \begin{array}{c} n \ r \end{array} \right) \), it represents the number of ways to choose "r" items from "n" without caring about the order.

• For instance, choosing 2 fruits from a basket containing apples, oranges, and bananas can result in combinations such as apple-banana, apple-orange, and banana-orange.
• Thus, mathematically, it employs the expression: \( \frac{n!}{r!(n-r)!} \).

Understanding this mathematical tool allows us to solve complex problems involving probabilities and distributions.

Mathematical Expressions

Mathematical expressions consist of combinations of numbers, variables, and mathematical operators. They are the fundamental language of mathematics, used to describe equations, functions, and formulas.
In the context of the provided exercise, the mathematical expression is given as a series of binomial coefficients:

• \( \left( \begin{array}{l}5 \ 0\end{array} \right) - \left( \begin{array}{l}5 \ 1\end{array} \right) + \left( \begin{array}{l}5 \ 2\end{array} \right) - \left( \begin{array}{l}5 \ 3\end{array} \right) + \left( \begin{array}{l}5 \ 4\end{array} \right) - \left( \begin{array}{l}5 \ 5\end{array} \right) \)
• This expression involves both simple arithmetic operations (addition and subtraction) and the calculation of combinations (binomial coefficients).

Each coefficient is calculated and then inserted back into the expression to find the final result, showcasing how these components work together to solve a complex problem.