Problem 20 - 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 expression evaluates to 0.

Step 1: Evaluate Binomial Coefficients

First, compute each of the binomial coefficients used in the expression:- \( \binom{5}{0} = 1 \)- \( \binom{5}{1} = 5 \)- \( \binom{5}{2} = 10 \)- \( \binom{5}{3} = 10 \)- \( \binom{5}{4} = 5 \)- \( \binom{5}{5} = 1 \)

Step 2: Substitute Binomial Coefficients into Expression

Now substitute the binomial coefficients into the expression:\[1-5+10-10+5-1\]

Step 3: Simplify the Expression

Perform the arithmetic operations to simplify the expression:1. \(1 - 5 = -4\)2. \(-4 + 10 = 6\)3. \(6 - 10 = -4\)4. \(-4 + 5 = 1\)5. \(1 - 1 = 0\)

Step 4: Conclusion: Final Answer

The value of the expression is 0.

Key Concepts

Binomial Coefficients

Binomial coefficients come from the binomial theorem and are used in algebra to express coefficients of terms in a binomial expansion. They are often denoted as \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. This is often read as \( n \) choose \( k \). The formula to calculate binomial coefficients is:

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

Where \( n! \) is the factorial of \( n \), which is the product of all positive integers up to \( n \).
For example, in the expression from the exercise, you have coefficients for \( n = 5 \) and varying \( k \). Calculating binomial coefficients helps form the terms of complex binomial expressions. Binomial coefficients follow certain patterns and can be easily found using Pascal's Triangle, which organizes them in triangular form for quick reference.

Arithmetic Operations

Arithmetic operations involve the basic mathematical calculations used to manipulate numbers. These are addition, subtraction, multiplication, and division. In this particular exercise, we mainly perform subtraction and addition.
To simplify the expression, you follow these steps:

• Apply operations on numbers from left to right.
• First subtract adjacent numbers and then add as per the order of expression.
• Keep track of positive and negative values as you move through the sequence.

For example, when you have the expression \( 1 - 5 + 10 - 10 + 5 - 1 \), you go step by step:- First compute \( 1 - 5 = -4 \), which gives a negative value.
- Next, add \(-4 + 10 = 6\). Now you have a positive value.
- Continue with \( 6 - 10 = -4 \), and so forth until you reach the end of the expression. Arithmetic operations are foundational in solving any algebraic expressions.

Expression Simplification

Expression simplification involves reducing a complex expression to its simplest form. This process consists of combining like terms and applying arithmetic operations.
In the case of the original problem, simplifying translates into sequentially executing arithmetic operations to arrive at a single number. Here are a few tips for simplification:

• Identify the operations given in the order of the expression.
• Always keep track of the signs (+ or -) impacting the results.
• Revisit each calculation step to ensure accuracy.

Simplifying \( 1 - 5 + 10 - 10 + 5 - 1 \) leads you to evaluate the operation step by step, carefully maintaining the order. Each step leads to fewer terms until you get a final result, which simplifies your understanding of how different parts of an expression interact. Becoming proficient in simplifying expressions is essential in algebra as it lays the groundwork for solving more complex mathematical problems.