Problem 9 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

a) Berechnen sie fur osk 55 alle Binamialkoeffizienten \(\left(\begin{array}{l}5 \\ k\end{array}\right)\), Vergleichen sie mit den zahlen in der fünften zeile des Pascalschen Dreiecks. b) Bestimmen sie \(\sum_{k=0}^{2}\left(\begin{array}{l}2 \\\ k\end{array}\right] a^{2-k} b^{k}\) und \(\sum_{k=0}^{3}\left(\begin{array}{l}3 \\\ k\end{array}\right) a^{3-k}_{b} k\).

Step-by-Step Solution

Answer

a) Binomial coefficients: 1, 5, 10, 10, 5, 1. b) Sums: \(a^2 + 2ab + b^2\), \(a^3 + 3a^2b + 3ab^2 + b^3\).

Step 1: Calculate Binomial Coefficients for \( \binom{5}{k} \)

The binomial coefficients \( \binom{5}{k} \) can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). We need to calculate for \( k = 0, 1, 2, 3, 4, 5 \):- \( \binom{5}{0} = 1 \)- \( \binom{5}{1} = 5 \)- \( \binom{5}{2} = 10 \)- \( \binom{5}{3} = 10 \)- \( \binom{5}{4} = 5 \)- \( \binom{5}{5} = 1 \)

Step 2: Compare with Pascal's Triangle

The fifth row of Pascal's Triangle (starting with row 0) is \( 1, 5, 10, 10, 5, 1 \). This matches the coefficients calculated: \( \binom{5}{0}, \binom{5}{1}, \binom{5}{2}, \binom{5}{3}, \binom{5}{4}, \binom{5}{5} \).

Step 3: Calculate the First Sum for \( \sum_{k=0}^{2}\binom{2}{k} a^{2-k} b^k \)

Calculate each term and sum:- \( k = 0: \binom{2}{0} a^2 b^0 = 1 \cdot a^2 = a^2 \)- \( k = 1: \binom{2}{1} a^{2-1} b^1 = 2ab \)- \( k = 2: \binom{2}{2} a^{2-2} b^2 = 1 \cdot b^2 = b^2 \)The sum is \( a^2 + 2ab + b^2 \). This is the expansion of \( (a+b)^2 \).

Step 4: Calculate the Second Sum for \( \sum_{k=0}^{3}\binom{3}{k} a^{3-k} b^k \)

Calculate each term and sum:- \( k = 0: \binom{3}{0} a^3 b^0 = 1 \cdot a^3 = a^3 \)- \( k = 1: \binom{3}{1} a^{3-1} b^1 = 3a^2b \)- \( k = 2: \binom{3}{2} a^{3-2} b^2 = 3ab^2 \)- \( k = 3: \binom{3}{3} a^{3-3} b^3 = b^3 \)The sum is \( a^3 + 3a^2b + 3ab^2 + b^3 \). This is the expansion of \( (a+b)^3 \).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it in the previous row. It's a fascinating mathematical tool because it generates the coefficients needed to expand binomial expressions. Each row corresponds to the coefficients of the expanded terms of \((a + b)^n\), where \( n \) is the row number, starting from the 0th row. For example:

• Row 0: 1 (corresponds to \((a + b)^0 = 1)\)
• Row 1: 1, 1 (corresponds to \((a + b)^1 = a + b)\)
• Row 2: 1, 2, 1 (corresponds to \((a + b)^2 = a^2 + 2ab + b^2)\)

Pascal's Triangle not only simplifies computations in algebra but also offers insights into various combinatorial problems, providing an easy way to calculate combinations like \(\binom{n}{k}\). In the original exercise, comparing the calculated binomial coefficients \(\binom{5}{k}\) with the 5th row of Pascal's Triangle helps to visually confirm the accuracy of the results.

Binomial Theorem

The Binomial Theorem provides a formula for expanding powers of binomials. According to this theorem, we can expand any power of a binomial as a sum involving binomial coefficients: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(\binom{n}{k}\) represents binomial coefficients, calculated using the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
In basic terms, the Binomial Theorem tells us how to distribute the power \(n\) across the terms \(a\) and \(b\) and how to efficiently compute each term in the expansion. Let's consider the expressions given in the problem:

• For \((a+b)^2\), it expands to \(a^2 + 2ab + b^2\), matching the calculated sum from \( \sum_{k=0}^{2} \binom{2}{k} a^{2-k} b^k \).
• For \((a+b)^3\), it expands to \(a^3 + 3a^2b + 3ab^2 + b^3\), which aligns perfectly with the sum \( \sum_{k=0}^{3} \binom{3}{k} a^{3-k} b^k \).

The Binomial Theorem is a powerful tool in algebra, providing a straightforward way to handle complex polynomial expansions. It's especially beneficial in studying probability, calculus, and other areas of advanced mathematics.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations, sequences, arrangements, and counting. Within this field, binomial coefficients play a significant role. They help express combinations, or the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to order.
When you encounter binomial coefficients, you're essentially solving a combinatorial problem. For instance, when we calculate \( \binom{5}{2} \), we determine the number of ways to select 2 items from a group of 5. In the original exercise, finding different binomial coefficients such as \( \binom{5}{k} \) is about counting combinations.

• \(\binom{5}{0}\) is the number of ways to choose none, which is 1.
• \(\binom{5}{1}\) is the number of ways to choose just 1, which is 5.
• \(\binom{5}{2}\) is the number of ways to choose two at a time, which is 10.

Combinatorics offers tools that are foundational in understanding probability, algorithms, and various other topics in computer science and mathematics. Each coefficient reflects possible selections, enabling calculations critical in diverse analytical and practical applications.

Sum of Binomial Series

The sum of a binomial series involves adding up the terms of a binomial expansion. This is what you do when you expand expressions like \( (a + b)^n \), using the Binomial Theorem we talked about earlier. To compute it, you used binomial coefficients:
In the original problem, two sums had to be calculated:

• \( \sum_{k=0}^{2} \binom{2}{k} a^{2-k} b^k = a^2 + 2ab + b^2\).
• \( \sum_{k=0}^{3} \binom{3}{k} a^{3-k} b^k = a^3 + 3a^2b + 3ab^2 + b^3\).

These sums actually represent the simplified expressions for \( (a+b)^2 \) and \( (a+b)^3 \).
The sum of a binomial series is significant because it reflects not only algebraic expansion but also showcases how polynomial growth occurs systematically. Each term in the series grows by integrating both variables \( a \) and \( b \) in structured combinations, underlining the elegance and coherence of mathematical patterns. It’s used widely in calculus, probability theories, and any mathematical aspect involving series expansions.