Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ _{4} C_{3} $$

Step-by-Step Solution

Answer

The expression \( _{4}C_{3} \) evaluates to 4.

Step 1: Understand the Combination Notation

The notation \( _{n}C_{r} \) refers to the number of combinations or ways to choose \( r \) objects from a set of \( n \) objects without regard to the order. It's calculated using the formula: \[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \] where \(!\) denotes factorial, which is the product of all positive integers up to a given number.

Step 2: Identify Values of n and r

In the expression \( _{4}C_{3} \), identify \( n = 4 \) and \( r = 3 \). These values are needed for substitution into the combination formula.

Step 3: Substitute Values into the Formula

Substitute \( n = 4 \) and \( r = 3 \) into the formula:\[ _{4}C_{3} = \frac{4!}{3!(4-3)!} \] which simplifies to:\[ \frac{4!}{3! \cdot 1!} \]

Step 4: Calculate Factorials Required

Calculate the factorials you need:- \(4! = 4 \times 3 \times 2 \times 1 = 24\)- \(3! = 3 \times 2 \times 1 = 6\)- \(1! = 1\)Substitute these back into the expression: \[ \frac{24}{6 \cdot 1} \]

Step 5: Simplify the Expression

Simplify the fraction by dividing the numerator by the denominator:\[ \frac{24}{6} = 4 \] Thus, the number of combinations is 4.

Key Concepts

Factorial

The concept of a factorial is fundamental in mathematics, and it's denoted with an exclamation mark, such as in the expression "3!". The factorial of a number is the product of all whole numbers from 1 up to that number. It is an important concept when dealing with permutations, combinations, and other areas of mathematics.

• For example, the factorial of 4 (written as 4!) is calculated as follows: 4 × 3 × 2 × 1 = 24.
• The factorial of 3, on the other hand, is 3 × 2 × 1 = 6.
• It's important to remember that the factorial of 0 is defined as 1.

Factorials allow us to determine the number of ways to arrange objects. They are crucial for calculating permutations and combinations, which in turn are used in various fields such as statistics and probability theory. By understanding the basic operation of factorials, students can easily apply them in more complex formulas, like those involving combinations.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and selecting items. It's all about finding ways to count things when the order does not matter. Typically used in probability and statistics, combinatorics involves concepts such as factorials, permutations, and combinations.

• One fundamental goal of combinatorics is to determine how many ways certain events can happen without having to list all possible outcomes.
• For example, combinatorics helps to calculate lottery odds, seating arrangements, and creating different groups from a set.

To grasp combinatorics, students need familiarity with some basic principles: these include understanding combinations which detail how many different ways you can choose items without caring about their order and permutations which account for order.
In the context of this exercise, the focus is on combinations. Understanding the difference between combinations and permutations is crucial, as permutations account for order while combinations do not.

Binomial Coefficient

The binomial coefficient is a concept that signifies the number of ways to choose a certain number of elements from a larger set, disregarding the order of selection. It is symbolized as "_nC_r" and calculated using the formula:\[_{n}C_{r} = \frac{n!}{r!(n-r)!}\]The binomial coefficient plays a key role in combinatorics, especially in problems where you need to find out how many combinations are possible.

• In the expression \(_{4}C_{3}\), 4 is the total number of items and 3 is the number of items to choose.
• To identify \(_{4}C_{3}\), substitute into the formula: \(\frac{4!}{3!(4-3)!}\). This results in \(\frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 1}\), simplifying to 4.
• The result, 4, indicates there are four ways to choose 3 items from a set of 4.

Binomial coefficients are generally used in binomial theorems and solving combinatorial problems. They provide insight into possible arrangements or selections in various fields such as algebra and probability.