Problem 14 - College Algebra

Question

In Exercises \(9-16,\) use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{4} C_{4} $$

Step-by-Step Solution

Answer

\(_4C_4 = 1.\)

Step 1: Identifying n and r

In the given problem, \(_{4} C_{4}\), n equals to 4 (the total number of items) and r equals to 4 (the number of items to choose). So, we have n = 4 and r = 4.

Step 2: Applying the combination formula

Apply the combination formula \(_{n} C_{r} = \frac{n!}{r!(n-r)!}\). Substituting n = 4 and r = 4 in this formula we get \(_4C_4 = \frac{4!}{4!(4-4)!}\).

Step 3: Computing the factorial values

Calculate the factorial values. 4! equals to 4*3*2*1 = 24. And (4-4)! or 0! equals to 1 (as the factorial of 0 is defined to be 1).

Step 4: Final calculation

Substitute the factorial values back into the formula, which gets us \(_4C_4 = \frac{24}{24*1}.\) Simplifying this gives us \(_4C_4 = 1.\)

Key Concepts

Factorial

A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a given number.
For instance, the factorial of 4, written as 4!, is calculated by multiplying all whole numbers from 1 to 4 together:

• 4! = 4 × 3 × 2 × 1 = 24.

Factorials are essential in combinatorics and other areas of mathematics because they help in calculating permutations and combinations.
A key thing to remember is that the factorial of 0, written as 0!, is defined to be 1.
This might seem a bit strange initially, but defining 0! as 1 helps maintain consistent calculations across mathematical formulas.

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics, often used to find the number of ways to choose a subset of items from a larger set.
It is denoted as \(_{n}C_{r}\), which means the number of combinations of \(r\) items from a set of \(n\) items.
The formula for the binomial coefficient is:

• \(_{n}C_{r} = \frac{n!}{r!(n-r)!}\)

This formula takes into account the factorial of the total number of items \(n!\), divided by the factorial of the number of items chosen \(r!\).
Don't forget to also divide by the factorial of the number of items not chosen \((n-r)!\).
This formula allows you to calculate the total possible combinations without considering the order of selection.

Combinatorial Mathematics

Combinatorial mathematics is the study of counting, both as a means and an end in obtaining results, as well as certain properties of finite structures.
A primary application of combinatorial mathematics is the calculation of combinations and permutations to solve problems related to grouping or arranging items.
When you hear the term combinations in this field, it typically refers to selecting items from a group where the order does not matter, as opposed to permutations, where the order is important.
Combinatorial mathematics is not just limited to arithmetic counting.
It often involves deeper levels of analysis to solve complex problems in various fields, from computer science to probability theory.