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 evaluated expression equals 32.

Step 1: Understand the Expression

The expression consists of a sum of binomial coefficients: \( \binom{5}{0} + \binom{5}{1} + \binom{5}{2} + \binom{5}{3} + \binom{5}{4} + \binom{5}{5} \). These represent the number of ways to choose \( k \) items from 5, and they are part of the expansion of \((1+x)^n\).

Step 2: Evaluate Individual Binomial Coefficients

Calculate the value of each binomial coefficient in the sum:- \( \binom{5}{0} = 1 \)- \( \binom{5}{1} = 5 \)- \( \binom{5}{2} = 10 \)- \( \binom{5}{3} = 10 \)- \( \binom{5}{4} = 5 \)- \( \binom{5}{5} = 1 \)

Step 3: Sum the Values of the Binomial Coefficients

Add all the values calculated for the binomial coefficients:\[ 1 + 5 + 10 + 10 + 5 + 1 = 32 \]

Step 4: Conclusion

The sum of the binomial coefficients \( \binom{5}{0} + \binom{5}{1} + \binom{5}{2} + \binom{5}{3} + \binom{5}{4} + \binom{5}{5} \) is 32.

Key Concepts

Binomial Coefficients

Binomial coefficients are a key component in combinatorics. They are numbers that appear in the binomial theorem and are often denoted as \(\binom{n}{k}\), which reads as "n choose k." This notation represents the number of ways to choose \(k\) items from \(n\) items without regard to order. These coefficients are calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \(n!\) (factorial of \(n\)) is the product of all positive integers up to \(n\). For example, \(\binom{5}{2}\) is calculated as \(\frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\).
Binomial coefficients have several properties that make them particularly useful.

• Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\).
• Summation: Adding all the coefficients of a row in Pascal's Triangle equals \(2^n\).

Combinatorics

Combinatorics is the branch of mathematics that deals with counting and arrangement of objects. It explores the different ways in which certain arrangements can occur.
One of its primary goals is to find a systematic method for counting the number of possible configurations. In the context of our exercise, combinatorics explains the logic behind choosing subsets of items. When you calculate a binomial coefficient, you're engaging in a combinatorial process. A common situation in which you might use combinatorics is determining how many different groups of people can sit in a room of chairs or how many different handshakes can occur among a group of people. Combinatorics also involves permutations, which consider the order of arrangement, unlike combinations, where the order does not matter. Understanding combinatorics helps in simplifying problem-solving in complex counting problems, ensuring that each potential arrangement or selection is considered effectively.

Polynomial Expansion

Polynomial expansion is the process of expressing a power of a binomial as a sum of terms involving binomial coefficients. The binomial theorem provides a formula for this expansion. When you raise a binomial \((a + b)\) to a power \(n\), it can be expanded as:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Each term in the expansion represents a particular arrangement of multiplying the terms \(a\) and \(b\). The binomial coefficient \(\binom{n}{k}\) in the expansion determines the number of ways to select \(k\) occurrences of \(b\) and \(n-k\) occurrences of \(a\).
A practical example is expanding \((1+x)^5\). The expansion includes terms such as \(\binom{5}{2} x^2\), indicating combinations of terms from the binomial.Polynomial expansions are fundamental in algebra and are heavily used in calculus and statistics, where they simplify the calculation of probabilities and evaluate functions.