Question

Prove \(\sum_{k=0}^{n} C(n, k)^{2}=C(2 n, n)\) (a) Prove the identity using the fact that \((1+x)^{2 n}=(1+x)^{n}(1+x)^{n}\), (b) Give a combinatorial proof of the identity in part a. (c) Find the number of 14 -digit binary sequences for which the number of 1 's in the first seven digits is the same as the number of 0 's in the last seven digits of the sequence. Enumerate all such sequences of length six.

Short answer

\(\sum_{k=0}^{n} C(n, k)^2 = C(2n, n)\). 14-digit sequence has 858 such, 6-digit 6 possibilities.

Step-by-step solution

Expand Using Binomial Theorem

First, use the binomial theorem to expand \[(1+x)^{2n} = \sum_{k=0}^{2n} C(2n, k)x^k\].This gives us a general formula for each coefficient when raised to a power.

Separate the Powers

Express \[(1+x)^{2n} = (1+x)^n(1+x)^n\].When expanded, this becomes\[(\sum_{i=0}^{n} C(n, i) x^i)(\sum_{j=0}^{n} C(n, j) x^j) = \sum_{k=0}^{2n} \left( \sum_{i+j=k} C(n, i)C(n, j) \right) x^k\].

Set Coefficients Equal

By comparing coefficients from Step 1 and Step 2, \[ C(2n, n) \] must equal \[ \sum_{i+j=n} C(n, i)C(n, j) \].This gives us \(\sum_{k=0}^{n} C(n, k)^2 = C(2n, n)\) since the sum of products of binomial coefficients must match the binomial coefficient for \(x^n\).

Combinatorial Interpretation

Consider choosing \(n\) people from \(2n\) distinct people. This is represented by \(C(2n, n)\). Alternatively, split the \(2n\) people into two groups (\(n\) each) and count the ways \(k\) people from group one and \(n-k\) from group two can form a team of \(n\). This is equivalent to \(\sum_{k=0}^{n} C(n, k)^2\).

Define the Binary Sequence Problem

Consider a sequence of 14 binary digits. For the sequence to have an equal number of 1s in the first seven digits and equal number of 0s in the last seven digits, each half must have the same number of 1s and 0s.

Enumerate Length-6 Sequences

For a sequences of length six, matching criteria are: 0 or 3 1s in the first three positions (and similarly for 0s in the last three). Possible sequences are: - 000111, - 001110, - 010101, - 011011, - 110001, - 111000.

Key concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that allows us to expand expressions of the form \((a + b)^n\). It's a way to find the coefficients of each term in this expanded polynomial. For example, for \((1 + x)^n\), the binomial theorem tells us the expansion is \(\sum_{k=0}^n C(n, k) x^k\), where \(C(n, k)\) are the binomial coefficients.
This theorem not only provides us with a method to expand polynomials but also connects deeply with combinatorics. Here, we used the binomial theorem to express \((1+x)^{2n}\) as \(\sum_{k=0}^{2n} C(2n, k)x^k\). This step is crucial for proving combinatorial identities. By expressing complex algebraic expressions through simpler binomials, the theorem bridges algebra and combinatorics.

• This expansion helps us understand the distribution of terms in a polynomial expansion.
• It establishes a connection between algebraic functions and combinatorial counting methods.
• The theorem is versatile in solving both theoretical and practical problems related to binomials.

Binomial Coefficients

Binomial coefficients, denoted as \(C(n, k)\) or sometimes \(\binom{n}{k}\), represent the coefficients of the terms in the binomial expansion. They are a way of counting how many ways there are to choose \(k\) elements from a set of \(n\) elements.
In simpler terms, these coefficients tell us how many combinations there are to pick \(k\) items out of \(n\). In our exercise, we used these coefficients to solve for specific identities. For any natural number \(n\) and \(k\) less than or equal to \(n\), these coefficients are calculated using\[ C(n, k) = \frac{n!}{k!(n-k)!} \]where \(!\) denotes a factorial, the product of an integer and all the integers below it.

• They give a clear numerical way to approach problems involving combinations.
• Binomial coefficients are fundamental in combinatorics for counting combinations.
• They are used extensively in probability, algebra, and discrete mathematics.

Combinatorial Identities

Combinatorial identities are equations involving binomial coefficients where both sides represent the same combinatorial object or set of outcomes. The identity we proved involves showing that both sides represent the same way to choose \(n\) elements from \(2n\) elements.
In our exercise, the identity \(\sum_{k=0}^{n} C(n, k)^2 = C(2n, n)\) was simplified by connecting algebraic manipulations to combinatorial reasoning. Combinatorial proofs often assign a real-world scenario to the variables involved, making it easier to grasp the solution. Here, \(C(2n, n)\) denotes the number of ways to choose \(n\) people from a group of \(2n\) people.
Alternatively, \(\sum_{k=0}^{n} C(n, k)^2\) calculates the number of ways to split the group into two subsets then choose \(k\) members from the first and \(n-k\) from the other to complete the team.

• They provide a deeper understanding of the relationships within combinatorial figures.
• Combinatorial identities simplify complex problems by revealing equivalences in counting methods.
• They bridge disciplines, merging algebraic methods with practical combinatorial counting ways.