Problem 14 - Algebra and Trigonometry

Question

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{4} \mathrm{C}_{4}\)

Step-by-Step Solution

Answer

The value of \(_{4} C_{4}\) is 1. This means there is 1 way to select 4 items from a set of 4.

Step 1: Identification of Variables

In this case, our \(n\) and \(r\) values both equal to 4 because \(_4 C_4)\.

Step 2: Use the binomial coefficient formula

Substituting \(n = 4\) and \(r = 4\) into the formula \(_n C_r = \frac {n!}{r!(n-r)!}\), we get \(_4 C_4 = \frac {4!}{4!(4-4)!}\). Here, '!' is the factorial function, where \(n! = n*(n-1)*(n-2)....*3*2*1\) and \(0! = 1\) by definition.\n So, \(_4 C_4 = \frac {4*3*2*1}{4*3*2*1*1} = 1

Key Concepts

Binomial Coefficient

The binomial coefficient is a very interesting concept in combinatorics, often represented as \( _{n}C_{r} \) or sometimes \( \binom{n}{r} \). This is used to calculate the number of ways we can choose \( r \) items from a total of \( n \) items. It's a crucial part of understanding combinations, where order does not matter, unlike permutations where the order is significant.

• The formula for calculating the binomial coefficient is given by:\[ _{n}C_{r} = \frac{n!}{r!(n-r)!} \]
• In this formula, \( n! \) and \( r! \) are factorials, which we will delve into in the next section.
• Importantly, the binomial coefficient highlights symmetries as \( _{n}C_{r} = _{n}C_{n-r} \). This shows that choosing \( r \) items from \( n \) is equivalent to leaving \( n-r \) items behind.

When applying this to our example, both \( n \) and \( r \) are equal to 4. Hence, \( _{4}C_{4} \) denotes choosing all 4 items from a group of 4, which logically is only possible in exactly one way. Therefore, \( _{4}C_{4} = 1 \). Understanding binomial coefficients can greatly simplify solving problems related to combinations in various fields such as probability and statistics.

Factorial Function

The factorial function is denoted by an exclamation mark (!), and it's a key mathematical operation used frequently in permutations and combinations. It is used to define the number of ways to arrange a set number of distinct items.

• The formal definition is: \( n! = n \times (n-1) \times (n-2) \cdots \times 2 \times 1 \).
• An important convention is that \( 0! = 1 \), even though 0 doesn't naturally fit into the pattern. This assignment makes the formula for combinations and other mathematical formulas work seamlessly.
• Factorials increase incredibly quickly with large \( n \), which is why many calculators and computers have a built-in factorial function.

For our exercise example, \( 4! \) equals \( 4 \times 3 \times 2 \times 1 = 24 \). When used in combination formulas like \( _{n}C_{r} \), they help determine how many different ways you can select items from a larger pool, factoring into the numerator and denominator components of the binomial coefficient formula.Factorials are essential in simplifying complex calculations, especially in probability and statistics.

Mathematical Expressions

Mathematical expressions are combinations of symbols and operations that represent a numerical value or relationship. Comprehending how to interpret these is vital in solving mathematical problems effectively.

• In our case, we are focusing on expressions involving the binomial coefficient and factorials, represented by \( _{n}C_{r} \) and \( n! \) respectively.
• An expression like \( _{4}C_{4} = \frac{4!}{4!(4-4)!} \) is simplified to \( \frac{24}{24 \times 1} = 1 \), showing using factorial division how combinations simplify.
• Working through mathematical expressions often requires recognizing patterns and applying rules systematically, which involves substituting known values and simplifying step by step as shown in the solution.

A firm grasp of expressions allows you to break down complex problems into manageable calculations. They form the foundation of translating real-world problems into solvable mathematical formulas and are pivotal in all areas of math, from algebra to calculus.