Problem 59
Question
Prove that for any positive integer \(n\) $$2^{n}=\left(\begin{array}{l}n \\\0\end{array}\right)+\left(\begin{array}{l}n \\\1\end{array}\right)+\left(\begin{array}{l}n \\\2\end{array}\right)+\cdots \\#\left(\begin{array}{l}n \\\n\end{array}\right)$$ \([\text {Hint: } 2=1+1 .]\)
Step-by-Step Solution
Verified Answer
Question: Prove that for any positive integer "n", the following combinatorial identity holds:
\(2^{n} = \left(\begin{array}{l}n \\0\end{array}\right) + \left(\begin{array}{l}n \\1\end{array}\right) + \left(\begin{array}{l}n \\2\end{array}\right) +\cdots+ \left(\begin{array}{l}n \\n\end{array}\right)\)
Answer: Use the binomial theorem with a=b=1 and simplify the right hand side of the equation to prove the given combinatorial identity for any positive integer \(n\).
1Step 1: Binomial theorem
Use the binomial theorem with a=b=1:
\((1 + 1)^n = \sum_{k=0}^n \left(\begin{array}{l}n \\k\end{array}\right) 1^{n-k} 1^k\)
2Step 2: Simplify the equation
Since 1 raised to any power is always 1, we can simplify the right hand side of the equation:
\(2^n = \sum_{k=0}^n \left(\begin{array}{l}n \\k\end{array}\right)\)
3Step 3: Rewrite the sum notation
Rewrite the sum notation as a combinatorial sum:
\(2^n = \left(\begin{array}{l}n \\0\end{array}\right) + \left(\begin{array}{l}n \\1\end{array}\right) + \left(\begin{array}{l}n \\2\end{array}\right) +\cdots+ \left(\begin{array}{l}n \\n\end{array}\right)\)
This proves the given combinatorial identity for any positive integer \(n\).
Key Concepts
Combinatorial IdentityExponentiationSummation Notation
Combinatorial Identity
In mathematics, a combinatorial identity is an equality involving combinatorial expressions. These are expressions that count the number of ways certain configurations can be arranged. The most common example is the binomial coefficient, symbolized by \( \binom{n}{k} \), which counts the number of ways to choose a subset of \(k\) elements from a larger set of \(n\) distinct elements.
The binomial theorem is a powerful tool in combinatorics that relates powers of a binomial to these coefficients. It shows that for any numbers \(a\) and \(b\), and any non-negative integer \(n\), the expansion of \( (a + b)^n \) can be expressed as a sum of terms involving these binomial coefficients:
Understanding these connections furthers your ability to solve a variety of mathematical problems beyond mere exponentiation, as combinatorial identities are essential in probability, statistics, and many areas of discrete mathematics.
The binomial theorem is a powerful tool in combinatorics that relates powers of a binomial to these coefficients. It shows that for any numbers \(a\) and \(b\), and any non-negative integer \(n\), the expansion of \( (a + b)^n \) can be expressed as a sum of terms involving these binomial coefficients:
\[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \]This formula explains not just how to expand binomial expressions, but also establishes a link between algebra and combinatorial counting. The exercise you are working on utilizes this theorem to prove a combinatorial identity by setting \(a = b = 1\), which yields a sum of binomial coefficients equal to \(2^n\).
Understanding these connections furthers your ability to solve a variety of mathematical problems beyond mere exponentiation, as combinatorial identities are essential in probability, statistics, and many areas of discrete mathematics.
Exponentiation
The concept of exponentiation refers to the mathematical operation of raising one number, the base, to the power of another number, the exponent. It is a shorthand notation to express repeated multiplication.
For instance, \(2^n\) is the product of multiplying two by itself \(n\) times. This operation is fundamental in many areas of mathematics, including algebra, where it's often used to represent growth or decay processes in nature and finance.
Understanding exponentiation is crucial when working with the binomial theorem, as the theorem itself provides a way to expand expressions raised to a power. The equation \(2^n\) is significant in binary systems and computer science since binary operations revolve around powers of two. In our exercise, establishing the equivalence of \(2^n\) with a sum of binomial coefficients illuminates the underlying structure of binary expansion and provides insight into why this identity holds true for any positive integer \(n\).
For instance, \(2^n\) is the product of multiplying two by itself \(n\) times. This operation is fundamental in many areas of mathematics, including algebra, where it's often used to represent growth or decay processes in nature and finance.
Understanding exponentiation is crucial when working with the binomial theorem, as the theorem itself provides a way to expand expressions raised to a power. The equation \(2^n\) is significant in binary systems and computer science since binary operations revolve around powers of two. In our exercise, establishing the equivalence of \(2^n\) with a sum of binomial coefficients illuminates the underlying structure of binary expansion and provides insight into why this identity holds true for any positive integer \(n\).
Summation Notation
Summation notation, denoted by the sigma symbol \(\Sigma\), is a concise way to represent the sum of a sequence of terms. It allows for an efficient expression of long sums, which is essential in mathematics, particularly in statistics and calculus.
The notation consists of the sigma symbol followed by an expression representing the terms to be added up. Underneath the sigma, you'll often find the index of summation (commonly \(i\), \(j\), or \(k\)) and the value at which it starts. Above the sigma, the ending value is stated. The entire notation signifies that you should sum up all values of the expression as the index goes from the starting value to the ending value.
In the context of our exercise, the notation \[ \sum_{k=0}^n \binom{n}{k} \] represents the sum of the binomial coefficients from \(k = 0\) to \(k = n\). This is essential in understanding the binomial theorem, as it allows us to express the entire expansion of \((1 + 1)^n\) as a single, compact sum.
The notation consists of the sigma symbol followed by an expression representing the terms to be added up. Underneath the sigma, you'll often find the index of summation (commonly \(i\), \(j\), or \(k\)) and the value at which it starts. Above the sigma, the ending value is stated. The entire notation signifies that you should sum up all values of the expression as the index goes from the starting value to the ending value.
In the context of our exercise, the notation \[ \sum_{k=0}^n \binom{n}{k} \] represents the sum of the binomial coefficients from \(k = 0\) to \(k = n\). This is essential in understanding the binomial theorem, as it allows us to express the entire expansion of \((1 + 1)^n\) as a single, compact sum.
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