Problem 34 - Algebra and Trigonometry

Question

Find the fifth term in the expansion of \((a b-1)^{20}\)

Step-by-Step Solution

Answer

The fifth term is \(4845 (ab)^{16}\).

Step 1: Identify the Binomial Expansion Formula

The binomial expansion formula is given by \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). We will use this formula to expand \((ab - 1)^{20}\).

Step 2: Recognize Variables x and y

In the expression \((ab - 1)^{20}\), we identify \(x = ab\) and \(y = -1\) for substitution into the binomial expansion formula.

Step 3: Determine k for the Fifth Term

The term number in the binomial theorem is represented as \(k+1\), which implies for the fifth term, we need \(k = 5 - 1 = 4\).

Step 4: Apply Binomial Formula for the Fifth Term

Substitute into the binomial formula to find the fifth term: \(\binom{20}{4} (ab)^{20-4} (-1)^4\). Simplifying, this becomes \(\binom{20}{4} (ab)^{16} (1)\).

Step 5: Calculate the Binomial Coefficient

Compute \(\binom{20}{4}\), which is the number of ways to choose 4 items from 20. Formula: \(\binom{20}{4} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845\).

Step 6: Simplify the Fifth Term

Substitute the value of the binomial coefficient into the expression for the fifth term: \(\binom{20}{4} (ab)^{16} (1) = 4845 (ab)^{16}\).

Key Concepts

Binomial Theorem

The binomial theorem provides a way to expand expressions raised to a power. This is especially useful when dealing with expressions like \((x + y)^n\). It gives us a systematic method to find each term in the expansion without multiplying everything out manually. This theorem is represented mathematically as:

• \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

Here, \(n\) is the power or exponent, and \(x\) and \(y\) are the terms being raised to the power \(n\). The sum runs from \(k = 0\) to \(n\), which means every value of \(k\) gives us a new term in the expansion.
It's like having a recipe where you systematically combine ingredients (\(x\) and \(y\)) in different quantities to create every possible dish (term) for a specified number of guests (the power \(n\)). So, each term in this expansion includes a combination of these ingredients weighted by the binomial coefficient.

Binomial Coefficient

At the heart of the binomial theorem, we find the binomial coefficient. It's denoted by \(\binom{n}{k}\) and tells us how many ways we can choose \(k\) items out of \(n\) available options, without considering the order. This is crucial in calculating each term in the binomial expansion.
The formula to compute the binomial coefficient \(\binom{n}{k}\) is:

• \(\binom{n}{k} = \frac{n!}{k! (n-k)!}\)

Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1\).
In our problem, the calculation of \(\binom{20}{4}\) involves choosing 4 elements out of 20. When calculated, it gives the number 4845, indicating that there are 4845 ways to select these elements. This coefficient becomes the multiplier in front of the corresponding term in the expansion.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations of things. It's an essential background for understanding the binomial theorem and binomial coefficients. Essentially, it gives us the mathematical tools to count various arrangements and selections of items.
In the context of the binomial expansion, combinatorics helps us understand why we're calculating the number of ways to choose certain numbers of elements. By understanding the principle of combination, we apply it to find how many ways we can distribute our terms \(x\) and \(y\) in the expansion of \((x + y)^n\).
Think of combinatorics like handling different puzzle pieces. While working with a binomial expansion, you're essentially determining how to put these pieces together in different groups or orders to complete the entire picture of our expanded polynomial.