Problem 12 - 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. $$\left(\begin{array}{l}9 \\\7\end{array}\right)$$

Step-by-Step Solution

Answer

The binomial coefficient \( \binom{9}{7} \) equals 36.

Step 1: Understand the Binomial Coefficient

The expression \( \binom{n}{k} \) represents the binomial coefficient, which calculates the number of ways to choose \( k \) items from \( n \) items without regard to the order. It is calculated using the formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). For this problem, \( n = 9 \) and \( k = 7 \).

Step 2: Calculate the Factorials

Calculate the factorials needed in the formula. \( 9! \) (9 factorial) is the product of all positive integers from 1 to 9, \( 7! \) is the product of all positive integers from 1 to 7, and \( (9-7)! = 2! \) is the product of 1 and 2.

Step 3: Compute 9!

Find the value of \( 9! \): \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \).

Step 4: Compute 7!

Find the value of \( 7! \): \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \).

Step 5: Compute 2!

Find the value of \( 2! \): \( 2! = 2 \times 1 = 2 \).

Step 6: Apply the Binomial Formula

Substitute the factorial values into the binomial formula: \( \binom{9}{7} = \frac{9!}{7!(9-7)!} = \frac{362880}{5040 \times 2} \).

Step 7: Simplify the Expression

Perform the division: \( \frac{362880}{5040 \times 2} = \frac{362880}{10080} = 36 \). Thus, \( \binom{9}{7} = 36 \).

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, used especially in permutations and combinations. A factorial, represented by the symbol !, is the product of all positive integers up to a given number.
For example:

• The factorial of 5, written as 5!, is calculated as: \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\].
• Likewise, 0! is defined to be 1, serving as a base case in mathematical expressions.

To compute large numbers like 9!, you continue this multiplication pattern until you reach 1.
Factorials grow at a very fast rate, making them useful for calculating the number of ways to organize a number of items or to solve various counting problems in mathematics.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects.
One central idea in combinatorics is the binomial coefficient, represented by \(inom{n}{k}\).This coefficient answers the question: "How many ways can you choose \(k\) items from \(n\) items without considering order?"
Essentially, combinatorics helps us simplify problems involving arrangements and selections.

• In our exercise, the binomial coefficient \(inom{9}{7}\)indicates choosing 7 objects from 9, without replacement and without particular order.
• By substituting into the formula \(\frac{n!}{k!(n-k)!}\),combinatorics transforms a real-world problem into a mathematical expression.

Mathematical Proofs

Mathematical proofs are logical arguments that verify the truth of mathematical statements.
They are essential for ensuring the reliability of conclusions drawn from mathematical reasoning.
In the context of our binomial coefficient problem, each step forms a part of a proof:

• Understanding the concept of factorials and substituting their values correctly showcases the correct application of the binomial formula.
• The act of simplifying and arriving at the final solution, here 36, validates that the approach and calculations were accurate.

Proofs can be informal or formal, but the essence is to show that each link in the chain of reasoning is justified and valid. This ensures that mathematical solutions are universally accepted and can be used as foundations for further study.