Problem 19 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \left(\begin{array}{c}{100} \\ {98}\end{array}\right) $$

Step-by-Step Solution

Answer

The value of \( \binom{100}{98} \) is 495.

Step 1: Understand the Notation

The expression \( \binom{100}{98} \) is a binomial coefficient. It represents the number of ways to choose 98 items from a group of 100 different items. This is commonly pronounced as "100 choose 98."

Step 2: Use the Binomial Coefficient Formula

The binomial coefficient \( \binom{n}{k} \) is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\] For our expression, \( n = 100 \) and \( k = 98 \). Substitute these values into the formula.

Step 3: Simplify Using Symmetry

The binomial coefficient \( \binom{n}{k} \) is equal to \( \binom{n}{n-k} \). So, \( \binom{100}{98} = \binom{100}{2} \). This allows us to work with smaller numbers, making calculations easier.

Step 4: Calculate the Coefficient

Now calculate \( \binom{100}{2} \) using the formula:\[\binom{100}{2} = \frac{100!}{2!(100-2)!} = \frac{100 \times 99}{2 \times 1}\]Simplifying gives:\[\frac{100 \times 99}{2} = 495\]

Step 5: Conclude the Calculation

The final result of the binomial coefficient \( \binom{100}{98} \) is 495. Thus, there are 495 ways to choose 98 items from 100.

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics, often referred to as the science of counting. It deals with the arrangement and selection of objects, making it a key component in solving problems involving combinations and permutations.
In our exercise, we are dealing with a specific combinatorial problem: computing a binomial coefficient, which calculates the number of ways to select a subset of items from a larger set. For the expression \( \binom{100}{98} \), this concept is known as a combination, where order does not matter as opposed to permutations where order is important.

• Combinations are used when the sequence of selection doesn’t affect the outcome.
• They’re essential in scenarios like lottery computations, team selections, and more.

Understanding the basics of combinatorics helps make sense of why and how various formulae, such as the binomial coefficient formula, are applied to derive meaningful results.

Factorials

Factorials play a crucial role in combinatorics, often appearing in the formulas to calculate the number of possible arrangements of a set of items. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \).
For instance, to calculate the binomial coefficient \( \binom{100}{98} \), we make use of the factorials in the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
• This requires computing the factorial of 100, 98, and also that of \(n - k\), which is 2 in this scenario.

Factorials grow very fast, which is why simplifying the calculations by finding equal paths such as \( \binom{100}{98} = \binom{100}{2} \) can significantly ease the computational process. This symmetry is a major feature when dealing with factorials in combinatorics.

Algebra

Algebra serves as the foundational language for solving equations and simplifying expressions, playing an integral role in solving combinatorial problems. In the case of binomial coefficients, algebraic manipulation is key to simplifying and computing results efficiently.
For example, by recognizing that \( \binom{100}{98} \) is equivalent to \( \binom{100}{2} \), we used the algebraic principle of symmetry. This reduces the computational complexity significantly:

• Instead of calculating \( \frac{100!}{98! \times 2!} \), we compute \( \frac{100 \times 99}{2 \times 1} \), which is much simpler.
• Algebra supports combinatorics by providing techniques to simplify expressions and solve equations rapidly and accurately.

Understanding algebraic principles enhances our ability to tackle complex mathematical problems by breaking them down into more manageable parts.