Problem 14 - Discrete Mathematics with Applications

Question

Find the largest binomial coefficient in the expansion of each. $$(x+y)^{6}$$

Step-by-Step Solution

Answer

The largest binomial coefficient in the expansion of \((x+y)^{6}\) is \(\binom{6}{3} = 20\).

Step 1: Understanding Binomial Coefficients

Recall that in the expansion of the binomial expression \((x+y)^{n}\), the binomial coefficients are represented by the symbol \(\binom{n}{k}\), where \(n\) is the power and \(k\) is the position of the term in the expansion. These coefficients can also be determined using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(n!\) is the factorial of \(n\). In our case, we have \(n=6\).

Step 2: Observing the Middle Terms

In the expansion of \((x+y)^{6}\), there are 7 terms. We want to see if the middle terms have the same binomial coefficient. The middle terms are those with \(k=3\) and \(k=4\). So, let's calculate the binomial coefficients for these values of \(k\): \[ \binom{6}{3}=\frac{6!}{3!3!} = 20 \] \[ \binom{6}{4}=\frac{6!}{4!2!} = 15 \] The binomial coefficient \(\binom{6}{3}\) is larger than \(\binom{6}{4}\). So, the largest coefficient is in the middle term. There is no need to check further, as the coefficients will decrease as we move away from the middle term.

Step 3: Conclusion

The largest binomial coefficient in the expansion of \((x+y)^{6}\) is \(\binom{6}{3} = 20\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions raised to a power in the form \( (x+y)^n \). It tells us that \( (x+y)^n \) can be expanded into a sum of terms involving powers of \( x \) and \( y \) with binomial coefficients as coefficients. These coefficients reflect the number of ways to pick a subset of items without regard to order.

For instance, the general form of the Binomial Theorem for \( (x+y)^n \) is given by:
\[ (x+y)^n = \binom{n}{0}x^n y^0 + \binom{n}{1} x^{n-1}y^1 + \ldots + \binom{n}{n}x^0 y^n \]
This pattern continues until all terms are exhausted. With every successive term, the power of \( x \) decreases by 1 while the power of \( y \) increases by 1, each accompanied by the appropriate binomial coefficient.

Factorial Notation

Factorial notation is used frequently in mathematics, especially in combinatorics and probability. The factorial of a non-negative integer \( n \) is denoted by \( n! \) and is the product of all positive integers less than or equal to \( n \) (e.g., \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \)).

In the context of binomial coefficients, factorial notation plays a critical role. Consider the binomial coefficient \( \binom{n}{k} \) which uses the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Factorial notation simplifies the calculation of these coefficients, accounting for the different arrangements of a subset out of a larger set. The denominator's product of factorials accounts for the repeated counts of identical sets, allowing for correct combinatoric counting in the expansion terms.

Combinatorics in Algebra

Combinatorics, the mathematics of counting, plays a central role when dealing with binomial expansions in algebra. It is not just about finding a coefficient but understanding how many combinations of terms in a polynomial there are.

For example, combinatorics comes into play when we wish to determine the coefficient of a particular term in a polynomial expansion. This is because the binomial coefficient \( \binom{n}{k} \) associated with each term of an expanded binomial expression corresponds to the number of ways \( k \) successes can be chosen from \( n \) trials. So in essence, when we calculate a binomial coefficient, we are using combinatorial principles to count the number of combinations.

Symmetry in Binomial Expansion

One key characteristic of the binomial expansion is its symmetry. This means that the coefficients of terms equidistant from the start and end of the expansion are identical. Specifically, the \( k \)th term from the beginning and the \( k \)th term from the end have the same binomial coefficient, formally stated as \( \binom{n}{k} = \binom{n}{n-k} \).

This property directly follows from the symmetry in the formula for the binomial coefficients:\[\binom{n}{k} = \frac{n!}{k!(n-k)!} = \binom{n}{n-k}\]The symmetry in the binomial expansion often simplifies the computations involved in finding particular terms or solving for coefficients, making the process more efficient by realizing that one only needs to compute up to the midpoint of the expansion.