Problem 19 - Precalculus: Mathematical 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 expression evaluates to 32.

Step 1: Understanding Binomial Coefficient

The expression is a sum of binomial coefficients. Each term is indicated as \( \binom{5}{k} \) where \( k \) ranges from 0 to 5. A binomial coefficient \( \binom{n}{k} \) denotes the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order of selection.

Step 2: Calculating Each Binomial Term

Evaluate each of the binomial coefficients individually:- \( \binom{5}{0} = 1 \)- \( \binom{5}{1} = 5 \)- \( \binom{5}{2} = 10 \)- \( \binom{5}{3} = 10 \)- \( \binom{5}{4} = 5 \)- \( \binom{5}{5} = 1 \)

Step 3: Summing the Binomial Coefficients

Add the calculated values from each binomial coefficient term:\[ 1 + 5 + 10 + 10 + 5 + 1 = 32 \]

Step 4: Conclusion

The value of the expression, which is the sum of all the binomial coefficients from \( k=0 \) to \( k=5 \) for \( n=5 \), is 32. This is equal to \( 2^5 \), illustrating the Binomial Theorem.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a quick way to expand expressions raised to a power, like \((x + y)^n\). In simpler terms, it tells us how to calculate the powers of sums. The formula is written as:\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]Here, \(\binom{n}{k}\) represents the binomial coefficients, which count how many ways we can pick \(k\) items from \(n\) total items. This theorem is incredibly useful in mathematics for tackling complex algebra problems easily.

• Expansion: It breaks down expressions like \((a + b)^5\) into smaller, manageable terms.
• Application: Helps in probability, statistics, and even in computer science for algorithm analysis.

When we use it in questions like the original exercise, it showcases the relationship between the coefficients and powers of expressions, illustrating that the sum of coefficients is always a power of two, specifically \(2^n\). This is why evaluating the sum of binomial coefficients like in the exercise gives us \(2^5 = 32\).

Combinatorics

Combinatorics is a field of mathematics dealing with combinations, permutations, and counting. It's like the mathematics of choice. It finds application in real life like organizing tournaments, decision-making processes, and algorithm design. One crucial concept in combinatorics is the binomial coefficient, symbolized as \(\binom{n}{k}\).

• Definition: Combinatorics analyzes the arrangements of sets and their subsets.
• Applications: Used in probability theory, for example, calculating the chances of drawing a royal flush in poker.

In the context of the original exercise, combinatorics helps evaluate how the sum of selecting different combinations of 5 elements (ranging from 0 to 5) yields the total possible subsets. Understanding this concept allows solving problems involving counting or arranging objects without manually listing every possibility. This skill is crucial for efficiency in mathematics and computing, aiding in developing algorithms and calculating probabilities.

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers showing binomial coefficients and providing a quick reference to these values. Each row in Pascal’s Triangle represents the coefficients of an expanded binomial expression. It begins with 1 at the top and each subsequent row ends with 1, with numbers in between formed by adding the two directly above it.

• Structure: Easy visual representation of binomial coefficients.
• Properties: Symmetrical; each row sums to a power of 2.
• Applications: Can be used to solve problems related to binomial expansions and combinatorics.

For instance, the 5th row of Pascal's Triangle is 1, 5, 10, 10, 5, 1, which corresponds exactly to the coefficients found when expanding \((x + y)^5\). Aligning with our original exercise, these coefficients, when summed, give the same result calculated using the binomial theorem. Pascal's Triangle is a friendly tool for understanding both combinatorics concepts and binomial expansions in a simpler visual way.