Problem 15 - College Algebra

Question

In Exercises \(9-16,\) use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{5} C_{0} $$

Step-by-Step Solution

Answer

\(_{5} C_{0} = 1\

Step 1 - Identification

This is a simple combination calculation. We are asked to find \(_{5} C_{0}\). Here, \(n=5\) and \(r=0\). The value corresponds to the number of methods to choose 0 items from a set of 5.

Step 2 - Substitute into combination formula

To compute , plug the values of \(n\) and \(r\) into the combination formula: \(_{n} C_{r} = \frac{n!}{r!(n-r)!}\). So, we have _{5} C_{0} = \(\frac{5!}{0!(5-0)!}\).

Step 3 - Calculate factorial for each part

From step 2, the expression breaks down to \(\frac{5!}{0!5!}\). Here, 5! equals 5×4×3×2×1 = 120. Additionally, 0! is defined to be 1. Therefore, the expression now unfolds into \(\frac{120}{1×120}\).

Step 4 - Compute the values

From step 3, further calculation gives the denominator to be 120 and thus the expression is 120 / 120

Step 5 - Final Answer

So our final answer, 120 / 120 equals 1. Thus \(_{5} C_{0}\) equals 1.

Key Concepts

Factorial Notation

Factorial notation is a mathematical concept used to describe the product of an integer and all the positive integers below it. Symbolized by an exclamation point (!), the factorial of a non-negative integer 'n' is written as 'n!'. For instance, when we consider the number 5, the factorial notation of 5, denoted as '5!', would be the product of all positive integers from 1 to 5. Thus,
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120. \]
It's important to note that the factorial of 0 is defined to be 1, written as \(0! = 1\). This might seem counterintuitive at first, but is crucial for the definition of the binomial coefficients and is an established convention in mathematics. Factorial notation plays a fundamental role in various areas of mathematics, including algebra, combinatorics, and calculus. Understanding how to calculate factorials is essential for solving problems involving permutations and combinations.

Binomial Coefficients

Binomial coefficients are integral components in the study of combinatorics. They represent the coefficients in the expansion of a binomial expression raised to a certain power. For instance, in the expression \((a + b)^n\), the coefficients of the terms are binomial coefficients. These coefficients can also be understood as the number of ways to choose 'r' elements out of a set of 'n' distinct elements without considering the order of selection, which is represented as \(_nC_r\) or \(\binom{n}{r}\).
The calculation of binomial coefficients utilizes factorial notation, given by the formula:
\[_nC_r = \frac{n!}{r!(n-r)!}\]
For example, using this formula we can find the binomial coefficient for choosing 0 items out of 5, denoted as \(_5C_0\), which, as our step-by-step solution shows, equates to 1. This demonstrates a fundamental rule in combinatorics stating that there is exactly one way to choose nothing from a set, hence \(_nC_0 = 1\) for any integer 'n'. In combinatorics, binomial coefficients are not only used in theoretical problems but also have real-world applications, like calculating probabilities.

Permutations and Combinations

In the realm of algebra and probability, permutations and combinations are concepts that deal with the arrangement of objects. Permutations are the different ways in which a set of objects can be arranged in sequence, accounting for the order of the objects. In contrast, combinations refer to the selection of objects from a set where the order does not matter.
To differentiate, consider a scenario where we have three books A, B, and C. The permutations would be all the different ways we can arrange these books in a row, such as ABC, ACB, BAC, BCA, CAB, and CBA. If we're looking at combinations, we would only be concerned with the selection of books, not the order. Hence, whether we choose AC or CA, it's considered one combination.
Mathematically, the number of permutations of 'n' objects taken 'r' at a time is expressed using 'nPr', calculated by the formula:
\[nPr = \frac{n!}{(n-r)!}\]
On the other hand, combinations are calculated using 'nCr' or \(_nC_r\) as we have seen with binomial coefficients. The relationship between permutations and combinations reflects the intrinsic order of the elements considered. Understanding these concepts is significant in solving problems related to probability, statistics, and various selection and arrangement scenarios.