Problem 7 - College Algebra

Question

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{c}100 \\\2\end{array}\right)$$

Step-by-Step Solution

Answer

The binomial coefficient of '100 choose 2' is 4950.

Step 1: Understanding Factorial

A factorial is a function that multiplies a given number by every number below it until 1. The factorial of 0 is 1. For example, the factorial of 5 is \( 5! = 5 * 4 * 3 * 2 * 1 = 120 \).

Step 2: Apply the Binomial Coefficient Formula

Once the concept of factorial is clear, it is time to apply the binomial coefficient formula \( \frac{n!}{k!(n-k)!} \), substituting the given values, n = 100 and k = 2. The binomial coefficient thus becomes \( \frac{100!}{2!(100-2)!} \).

Step 3: Simplify the Binomial Coefficient Expression

Simplify the binomial coefficient expression by calculating the values of the factorials. As 2! = 2 * 1 = 2, and (100-2)! is 98! while 100! = 100 * 99 * 98!, the expression \( \frac{100!}{2!(100-2)!} \) simplifies to \( \frac{100 * 99 * 98!}{2 * 1 * 98!} \). As the 98! terms cancel out, the expression further simplifies to \( \frac{100 * 99}{2} \).

Step 4: Evaluate the Binomial Coefficient

Finally, evaluate the expression \( \frac{100 * 99}{2} \). This simplifies to 4950.

Key Concepts

Factorial

The concept of factorials is fundamental in understanding various mathematical and statistical problems. A factorial, denoted by an exclamation mark (!), is the product of a positive integer and all the positive integers below it.
For example:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 3! = 3 × 2 × 1 = 6

The factorial of any number n, expressed as n!, involves multiplying n by all the integers less than it down to 1. Notably, the factorial of 0 is defined as 1, which is a special case to maintain consistency in mathematical equations.
Factorials grow very fast as numbers increase, which is why they play a crucial role in permutations and combinations.

Combinatorics

Combinatorics is the branch of mathematics dealing with the counting, arrangement, and combination of objects. This field answers the critical question: "How many ways can we arrange or select items?"
It involves:

• Permutations, which relate to the arrangement of items in a specific order.
• Combinations, concerning the selection of items where order doesn’t matter.

The binomial coefficient is a key concept in combinatorics, as it helps count the number of combinations of a certain number of elements. In essence, it provides a way of selecting k elements from a set of n elements, and it's calculated using factorials.

Binomial Theorem

The binomial theorem provides a formula for the expansion of binomials raised to any positive integer power. It states that:\[ (a + b)^n = \sum_{k=0}^{n} {\left(\begin{array}{c} n \ k \end{array}\right)} a^{n-k} b^k \]Here, the binomial coefficients \( \left(\begin{array}{c} n \ k \end{array}\right) \) are calculated using the factorial formula, and they indicate the number of ways to choose k elements from n elements.
The theorem is widely used in algebra, probability, and calculus to simplify polynomial expressions and solve complex problems with ease, leveraging the patterns of coefficients that naturally arise in expansions.

Mathematical Expression Simplification

Simplifying expressions is a core skill in mathematics that makes solving problems less cumbersome and more efficient. It involves reducing expressions to their simplest form without changing the value.
For binomial coefficients, simplification often involves:

• Cancelling common factors across numerator and denominator.
• Breaking down factorials into manageable multiplicative terms.

In our example, simplifying \( \frac{100!}{2!(100-2)!} \) to \( \frac{100 \times 99}{2} \) involves cancelling out \( 98! \) terms from the numerator and denominator, helping in finding the result more directly. Simplification not only streamlines calculations but also aids in understanding the mathematical relationships between components of expressions.