Problem 19 - Algebra and Trigonometry

Question

\(13-20=\) 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: Simplify Each Expression

To solve the exercise, we start by simplifying each binomial coefficient using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). We need to compute each of the following:\( \binom{5}{0} \), \( \binom{5}{1} \), \( \binom{5}{2} \), \( \binom{5}{3} \), \( \binom{5}{4} \), and \( \binom{5}{5} \).

Step 2: Calculate \( \binom{5}{0} \)

The expression is simplified as follows: \( \binom{5}{0} = \frac{5!}{0!(5-0)!} = \frac{5!}{1 \times 5!} = 1 \).

Step 3: Calculate \( \binom{5}{1} \)

The expression is simplified as follows: \( \binom{5}{1} = \frac{5!}{1!(5-1)!} = \frac{5!}{1 \times 4!} = 5 \).

Step 4: Calculate \( \binom{5}{2} \)

The expression is simplified as follows: \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \times 3!} = 10 \).

Step 5: Calculate \( \binom{5}{3} \)

The expression is simplified as follows: \( \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \times 2!} = 10 \).

Step 6: Calculate \( \binom{5}{4} \)

The expression is simplified as follows: \( \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \times 1!} = 5 \).

Step 7: Calculate \( \binom{5}{5} \)

The expression is simplified as follows: \( \binom{5}{5} = \frac{5!}{5!(5-5)!} = \frac{5!}{5! \times 1} = 1 \).

Step 8: Sum All the Values

Now, sum all the simplified values: \( 1 + 5 + 10 + 10 + 5 + 1 = 32 \). This is the value of the given expression.

Key Concepts

Combinatorics

Combinatorics is a fundamental part of mathematics focusing on counting, arranging, and combining items in sets. It provides tools for analyzing how different sets can be formed, often involving selection processes without replacement. In this context, binomial coefficients are a great example, as they determine how many different ways you can choose certain elements from a larger set.
Combinatorics often deals with problems like determining how many different groups can be formed under given constraints. For instance, in our exercise, we are evaluating \( inom{5}{k} \), which represents the number of ways to choose \( k \) elements from a set of 5. Each choice, from \( inom{5}{0} \) to \( inom{5}{5} \), highlights different combinations that can be achieved.

• Combinatorics helps in understanding the arrangement and selection of objects.
• Binomial coefficients are a part of combinatorial mathematics used to count combinations.
• Studying combinatorics enhances logical reasoning and problem-solving skills.

Understanding these concepts thoroughly can make more complex problems involving permutations and combinations easier to tackle.

Factorial Notation

Factorial notation is a compact way to represent the product of all positive integers up to a given number. For any integer \( n \), its factorial, denoted as \( n! \), is defined by \( n! = n imes (n-1) imes \ldots imes 1 \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are crucial in combinatorics, especially when dealing with permutations and combinations. They provide a way to calculate the total arrangements or sequences of sets.
The factorial operation plays a key role in the binomial coefficient formula, \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). Here, it determines how to divide the permutations by the number of ways to arrange selected items and unselected items. In steps 2 through 7 of our solution, we use factorials to simplify expressions like \( \binom{5}{2} = \frac{5!}{2!3!} \).

• Factorial notation helps calculate permutations and combinations.
• It is a fundamental concept for working with binomial coefficients.
• Understanding factorials aids in solving complex problems efficiently.

Mastering factorials simplifies working with various mathematical expressions that involve permutations and combinations.

Summation

Summation is the process of adding numbers or quantities to obtain their total. It is represented by the sigma notation (\( \Sigma \)). Summation is widely used in mathematics to express a series of terms succinctly. In the context of our exercise, summation aids in evaluating the total number of ways to choose elements from a set by combining individual binomial coefficients.
The exercise requires summing the binomial coefficients \( \binom{5}{0} + \binom{5}{1} + \binom{5}{2} + \binom{5}{3} + \binom{5}{4} + \binom{5}{5} \) to get a final result of 32. This result is part of a mathematical principle stating that for any integer \( n \), the sum of all \( \binom{n}{k} \) for \( k = 0 \) to \( n \) is \( 2^n \). So, \( 2^5 = 32 \) matches our expectation.

• Summation simplifies collecting terms for a final result.
• It helps in recognizing patterns and applying mathematical principles.
• Understanding summation assists in analyzing sequences and series.

Grasping the concept of summation is valuable for dealing with complex calculations and deriving concise solutions.