Problem 25 - Algebra 2

Question

Evaluate each expression. \(_{7} \mathrm{C}_{3}\)

Step-by-Step Solution

Answer

The value of \( _{7} \mathrm{C}_{3} \) is 35.

Step 1: Understanding the Combination Formula

The expression _{7} \(C\)_3 represents the number of ways to choose 3 items from 7, irrespective of the order, and is typically calculated using the combination formula. The combination formula for _{n} \(C\)_k is given by \( n! / (k!(n-k)!) \), where '!' denotes factorial of a number.

Step 2: Applying the Combination Formula

Apply the combination formula for _{7} \(C\)_3 with n = 7 and k = 3. Calculate the factorial for the numbers involved: \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), \(3! = 3 \times 2 \times 1\), and \(4! = 4 \times 3 \times 2 \times 1\). Now plug these values into the formula to get: \( \frac{7!}{3!(7-3)!} = \frac{7!}{3! \times 4!} \).

Step 3: Simplifying the Factorials

Cancel out the common terms in the numerator and the denominator. Since \(4! \)=\(4 \times 3 \times 2 \times 1\) is present in both the numerator (as part of \(7!\)) and denominator, we can simplify: \( \frac{7 \times 6 \times 5}{3 \times 2 \times 1} \).

Step 4: Calculating the Result

Calculate the result of the remaining expression by performing the multiplication and division: \( \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \).

Key Concepts

Combination Formula

When studying algebra and probability, you'll often come across the concept of making selections or combinations from a set without regard to the order of the elements. This is where the combination formula comes into play. It's a way to determine the number of possible combinations for selecting a subset of items from a larger set.

The formula is written as \( \_nC_k \) or \( C(n, k) \) where \( n \) is the total number of items to choose from, and \( k \) is the number of items to select. The formal definition of the combination formula is:
\[ \frac{n!}{k! \times (n - k)!} \]
Here, the exclamation point represents the factorial of a number, which takes us to our next foundational concept: the factorial.

Factorial

The factorial is a fundamental concept in mathematics, particularly in the fields of algebra, probability, and combinatorics. It is denoted by an exclamation point \( ! \) and represents the product of all positive integers up to a given number. For example, the factorial of 5, written as \( 5! \) is calculated as:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
It's important to note the special case of zero factorial, which is defined as \( 0! = 1 \). Factorials are key when dealing with permutations and combinations because they provide a way to calculate the total number of possible arrangements or selections.

Permutations and Combinations

Permutations and combinations are two related concepts that deal with the arrangement of items. Permutations account for the order of arrangement, meaning that the sequence in which items are arranged matters. In contrast, combinations do not consider the order - it's all about the selection of items regardless of how they are sequenced.

For instance, if you're selecting a committee of 3 members from a group of 10, combinations would be used because the order of selection doesn't matter. If, however, the order did matter—say, if you were assigning different roles to each member—permutations would be the appropriate calculation.

The significant difference in formulas is that permutations use:
\[ P(n, k) = \frac{n!}{(n-k)!} \]
While combinations, as previously discussed, utilize the combination formula without considering the different arrangements within the subset.

Algebraic Expressions

In the realm of algebra, an algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These expressions are the backbone of algebra and are used to represent various relationships and to solve problems. For example, the combination formula is an algebraic expression that involves both factorial notation and division.

Expressions can be simplified or manipulated according to the rules of algebra to solve for unknowns or to describe patterns and functions. Understanding how to work with algebraic expressions, including evaluating them and performing operations like factorials, is essential for tackling a wide array of mathematical problems.