Question

In Exercises 25-32, evaluate the expression. \({ }_5 C_1\)

Short answer

The value of \({ }_5 C_1\) is 5.

Step-by-step solution

Identify n and k

In this problem, \(n = 5\) which is the total number of items and \(k = 1\) which is the number of items to choose.

Use the formula

Replace \(n\) and \(k\) in the formula with 5 and 1 respectively. That gives \[C(5, 1) = \frac{5!}{1!(5-1)!}\].

Evaluate the factorial

Simplify \[C(5, 1) = \frac{5!}{1!(5-1)!}\] to get \[\frac{5 \times 4 \times 3 \times 2 \times 1}{1 \times (4 \times 3 \times 2 \times 1)}\].

Simplify the expression

When you simplify the expression, you get 5.

Key concepts

Understanding Factorials

A factorial, denoted with an exclamation mark (!), is a special mathematical function. It represents the product of an integer and all the positive integers below it. For example, the factorial of 5, written as 5!, equals 5 × 4 × 3 × 2 × 1 = 120. This sequence of multiplying down to 1 is what makes factorials unique and useful in many mathematical contexts.
Factorials are often used in permutations and combinations where arrangements and selections are needed. They simplify the calculations involved in choosing objects from a set. Importantly, 0! is defined to be 1. This might seem strange at first, but it makes the properties of factorials consistent, especially when dealing with probability and combinatorics.
When calculating a factorial in a combination formula, you typically end up simplifying the expression by canceling terms. This occurs because the numerator and denominator often have common factors, making the calculations more manageable.

Exploring Binomial Coefficient

The binomial coefficient is a key concept in combinatorics and is written as \( \binom{n}{k} \). It signifies the number of ways to choose k items from a set of n items without considering the order of selection. The formula to calculate it uses factorials:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

In the case of \( \binom{5}{1} \), we are calculating the number of ways to choose 1 item from a total of 5 items. Plugging into the formula gives:

• \( \frac{5!}{1!(5-1)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{1 \times 4 \times 3 \times 2 \times 1} \)

This simplifies to 5, showing that there are 5 different ways to select 1 item from 5. The use of factorials in the denominator (1! and (5-1)!) helps in achieving the correct count by canceling out extraneous permutations of the remaining items.

Introduction to Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of elements within a set. It provides tools to solve problems that involve selecting, arranging, or analyzing certain sets of objects. This field is widely applied in various disciplines, including computer science, physics, and statistics.
One of the primary applications of combinatorics is in calculating combinations, where order does not matter, and permutations, where order does matter. The binomial coefficient is a staple of combinatorial mathematics, producing results that easily solve complex selection processes.
An understanding of the basics of combinatorics, such as binomial coefficients and factorials, empowers solving advanced problems in graph theory, probability, and algorithm design. By mastering these concepts, you gain the ability to analyze and optimize choices and configurations efficiently.