Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. $$ C(4,2) $$

Step-by-Step Solution

Answer

There are 6 ways to choose 2 items from 4.

Step 1: Understand the Combinatorics Notation

The expression \( C(n, r) \) represents the number of combinations or ways to select \( r \) items from a set of \( n \) items without regard to the order of selection. It is also known as a binomial coefficient.

Step 2: Apply the Combination Formula

The formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \). In this exercise, \( n = 4 \) and \( r = 2 \). Therefore, substitute these values into the formula: \( C(4, 2) = \frac{4!}{2!(4 - 2)!} \).

Step 3: Calculate Factorials

Calculate each factorial needed for the formula. \( 4! = 4 \times 3 \times 2 \times 1 = 24 \), \( 2! = 2 \times 1 = 2 \), and \( (4-2)! = 2! = 2 \).

Step 4: Solve the Combination Expression

Substitute the calculated factorials back into the formula: \( C(4, 2) = \frac{24}{2 \times 2} = \frac{24}{4} \).

Step 5: Simplify the Result

Divide 24 by 4 to get the result: \( 6 \). Thus, there are 6 combinations of selecting 2 items from a set of 4.

Key Concepts

Binomial Coefficient

The binomial coefficient is an essential concept in probability and statistics, particularly in combinatorics. It is represented by the symbol \( C(n, r) \), and it tells us how many ways we can choose \( r \) items from \( n \) items without considering the order. When we say 'without considering order,' it means that the selection of items is unique, regardless of the sequence. For instance, selecting two items, A and B, from a set of four items is the same as selecting B and A — the sequence does not make a different combination.

• Notation: The binomial coefficient is also known as combinations or the combinatorics number.
• Example: Consider choosing 2 items from a set of 4. The notation is \( C(4, 2) \).

One common application is in binomial expansions, where these coefficients determine the weight of each term in the expanded form of \((a+b)^n\). Understanding how binomial coefficients work is crucial for solving many problems in mathematics.

Factorials

Factorials are mathematical expressions that are used, in part, to calculate how many ways things can be ordered or arranged. A factorial is written as \( n! \) and is calculated by multiplying all the integers from 1 to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

• Factorials grow very fast, which means as \( n \) increases, \( n! \) becomes very large very quickly.
• Factorials are integral in defining other mathematical expressions, such as permutations and combinations.

In the context of combinations, factorials are used to determine how many different ways \( r \) items can be selected from \( n \) items. The formula includes the factorials of \( n \), \( r \), and \((n-r)\). Understanding how to compute factorials is a foundational skill in solving combinatorial problems.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations, permutations, and the counting of objects that satisfy certain criteria. It plays a fundamental role in many fields such as computer science, statistics, and operations research.

• Combinations: Combinatorics often involves solving problems related to combinations, like in our exercise \( C(4, 2) \). It's about finding how many different selections can be made.
• Applications: Beyond mathematics, combinatorics is used in creating algorithms for optimization, understanding structures in complex networks, and solving logical puzzles.

The power of combinatorics lies in its ability to formalize counting and arrangement principles that might not be immediately intuitive. Using combinatorics, one learns how to break down complex problems into simpler parts, leading to solutions that are both efficient and elegant. As you practice, you'll see how impactful combinatorics is in solving real-world problems and theoretical questions alike.