Problem 14 - Precalculus with Limits

Question

In Exercises 5 - 14, calculate the binomial coefficient. \( \dbinom{100}{2} \)

Step-by-Step Solution

Answer

The binomial coefficient \( \dbinom{100}{2} \) is 4950.

Step 1: Understand the problem and identify its properties

The problem requires finding the binomial coefficient for two numbers, namely 100 and 2. This operation can be expressed mathematically as \( \dbinom{100}{2} \), which represents the number of ways to choose 2 items from 100.

Step 2: Apply the formula for binomial coefficients

The binomial coefficient is calculated using the formula \( \dbinom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n! \) represents the factorial of \( n \), \( k! \) the factorial of \( k \), and \( (n-k)! \) the factorial of \( n-k \). Applying this formula to our particular problem, we get \( \dbinom{100}{2} = \frac{100!}{2!(100-2)!} \).

Step 3: Calculate the factorials

First calculate the required factorials. Importantly, the factorial of 2 is \( 2*1 = 2 \) and the factorial of 98 is \( 98! \), a fairly large number.

Step 4: Simplify the expression

Before attempting to compute the large factorials, we can simplify the expression due to the properties of factorials. Specifically, \( 100! = 100*99*98! \). So we can rewrite the binomial coefficient as \( \dbinom{100}{2} = \frac{100*99*98!}{2*98!} \). The \( 98! \) terms cancel, leaving \( \dbinom{100}{2} = 100*99/2 \).

Step 5: Solve

Finally, execute the remaining calculation to get the result: \( \dbinom{100}{2} = 50*99 = 4950 \).

Key Concepts

Understanding Combinatorics

Combinatorics is an area of mathematics focused on counting, arrangement, and combination of elements within sets, often under certain constraints. When we calculate the number of possible selections or combinations of items from a larger set, we are venturing into combinatorial analysis. For instance, solving the exercise \( \dbinom{100}{2} \) involves combinatorics as we're finding out in how many ways we can choose 2 unique items from a set of 100.In real-life applications, this is akin to selecting 2 students to represent a class of 100, or picking 2 different fruits out of 100 in a basket. Remember, the order in which we pick them does not matter in this context, which is a defining feature of combinatorial problems that deal with combinations specifically.

Deciphering Factorial Notation

Factorial notation is used to describe the product of an integer and all the non-zero integers below it, which is an essential concept in various mathematical domains, including combinatorics. The notation \( n! \) represents the factorial of \( n \) and is defined only for non-negative integers. For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

When you encounter factorial notation in combinatorics, it's often in the context of evaluating the number of ways to order a set of elements. In the exercise provided, \( 100! \) signifies a very large number, being the product of all positive integers from 1 to 100. A helpful hint when dealing with factorials is to simplify expressions by canceling out common factorial terms, making computations more manageable, especially when they form part of a fraction as seen in binomial coefficients.

Mathematical Properties of Factorials

The properties of factorials play a crucial role in simplifying mathematical expressions, especially in solving combinatorial problems. One key property to recall is that multiplying or dividing factorials can lead to cancellation of common terms. This property was effectively utilized in the exercise when reducing \( \dbinom{100}{2} \) by cancelling \( 98! \) from the numerator and denominator.

An additional property worth mentioning is that the factorial of a number \( n \) is equal to \( n \) multiplied by the factorial of \( n-1 \) (i.e., \( n! = n \times (n-1)! \)). Such properties are fundamental when dealing with binomial coefficients, as they help to transform an otherwise daunting calculation into a straightforward multiplication or division task. Always remember \( 0! = 1 \) — a convention that serves as a base for defining the factorial function and aids in the computation of certain binomial coefficients where \( n = k \) or \( k = 0 \) without complicating the problem.