Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the following. In Exercises 17 and 18 , express the answer in terms of n. Do not use a calculator. $$\frac{7 !}{3 ! 4 !}$$

Step-by-Step Solution

Answer

The expression simplifies to 35.

Step 1: Understand the factorial

Factorial of a number, denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For example, \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).

Step 2: Substitute the factorial values

First, substitute the values of each factorial in the expression \( \frac{7!}{3! 4!} \). This gives:\[ \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1)(4 \times 3 \times 2 \times 1)} \]

Step 3: Cancel out common terms

Next, identify and cancel out common terms in the numerator and the denominator:- In the numerator \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\), you can cancel out \( 4 \times 3 \times 2 \times 1 \) because it matches with one of the denominator components (\( 4! \)).This simplifies the expression to:\[ \frac{7 \times 6 \times 5}{3 \times 2 \times 1} \]

Step 4: Simplify the expression further

Now simplify the expression by evaluating the remaining terms:- Numerator: \( 7 \times 6 \times 5 = 210 \)- Denominator: \( 3 \times 2 \times 1 = 6 \)- Division: \( \frac{210}{6} = 35 \)

Step 5: Final Answer in terms of \( n \)

Since there is no explicit dependency on \( n \) and the expression evaluated to a number, the result is the direct numeric outcome. Therefore:\[ \frac{7!}{3!4!} = 35 \]

Key Concepts

Combinatorics

Combinatorics is an area of mathematics that deals with counting and arranging objects. It's all about understanding how combinations and selections work in a set. For instance, if you have a pool of 7 people and you need to select 3 for a committee, combinatorics helps determine how many different ways this selection can be made.
In the exercise given, we're applying a part of combinatorics known as the binomial coefficient. The expression \(\frac{7!}{3!4!}\) represents the number of ways to choose 3 items from a set of 7, which is written as \(\binom{7}{3}\).

• \(7!\) computes the number of ways to arrange 7 items.
• \(3!\) accounts for the different arrangements of the 3 selected items.
• \(4!\) factors in the arrangement of the 4 items that are not being selected.

Thus, combinatorics allows us to simplify and systematically analyze these possibilities.

Permutations

Permutations are a crucial concept in combinatorics representing various ways to arrange or order a set of items. For example, if we have 3 items, there are 6 possible permutations or arrangements of these items, calculated via factorial as \(3! = 6\).
The given exercise isn't just about any permutation but rather determining specific ways to pick and arrange subsets, which ties into combinations, a related concept. When distinguishing between permutations and combinations, remember that permutations consider the order of arrangement, while combinations do not.
In problems like \(\frac{7!}{3!4!}\), permutations demonstrate how one would approach the full arrangement of 7 items (using \(7!\)) before simplifying it by accounting for internal arrangements (\(3!\) and \(4!\)).
Understanding permutations helps solidify your grasp on how factorials operate in sequential arrangements versus selection contexts.

Mathematical notation

Mathematical notation is the language of mathematics, providing a clear and universal way to express complex ideas, like the factorial operation denoted by the exclamation mark (\(!\)).
In this exercise, learning how factorial notation works is essential. When you see \(n!\), it implies the multiplication of all whole numbers less than or equal to \(n\). For instance, \(5!\) translates to \(5 \times 4 \times 3 \times 2 \times 1\).
Notation helps streamline problem-solving by providing concise options to express mathematical operations. In expressions like \(\frac{7!}{3!4!}\), knowing the function of factorials and division helps break down complex combinatorial problems into understandable parts.
By effectively using notation, you can write proofs, communicate solutions, and explore different mathematical fields more efficiently.