Problem 8 - College Algebra

Question

For the following exercises, evaluate the binomial coefficient. $$ \left(\begin{array}{l} 9 \\ 7 \end{array}\right) $$

Step-by-Step Solution

Answer

36

Step 1: Understand the Binomial Coefficient Notation

The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose \( k \) elements from \( n \) elements without regard to order. Mathematically, it can be calculated using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

Step 2: Identify the Values of n and k

In this problem, \( n = 9 \) and \( k = 7 \). This means we are looking to evaluate \( \binom{9}{7} \).

Step 3: Simplify the Formula

Recall that \( \binom{n}{k} = \binom{n}{n-k} \). Thus, \( \binom{9}{7} = \binom{9}{2} \), which reduces the problem to choosing 2 from 9.

Step 4: Calculate Factorials

Compute the necessary factorials: \( 9! = 362880 \), \( 2! = 2 \), and \( (9-2)! = 7! = 5040 \).

Step 5: Apply the Formula

Now compute \( \binom{9}{2} = \frac{9!}{2!7!} = \frac{362880}{2 \times 5040} \).

Step 6: Simplify the Expression

Simplify \( \frac{362880}{2 \times 5040} = \frac{362880}{10080} \).

Step 7: Perform the Division

Perform the division \( 362880 \div 10080 = 36 \). Therefore, \( \binom{9}{7} = 36 \).

Key Concepts

Factorials

A factorial, denoted by an exclamation mark (e.g., \( n! \)), is a special mathematical operation used frequently in counting and probability calculations. It represents the product of all positive integers less than or equal to a given number \( n \). For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Factorials are fundamental in the computation of binomial coefficients, which involve combinations of elements. In our exercise, calculating factorials was a vital step:

• \( 9! = 362880 \)
• \( 2! = 2 \)
• \( 7! = 5040 \)

Using these results, we simplified the problem’s calculations. Understanding factorials properly lays a good foundation for learning permutations and combinations.

Combinatorics

Combinatorics is the branch of mathematics dealing with the selection, arrangement, and combination of objects. It's at the heart of solving problems involving permutations and combinations.
Here’s why it’s important:

• It helps in calculating the number of possible combinations of a set of items without regard to order.
• The binomial coefficient \( \binom{n}{k} \) is a central concept in combinatorics, showing the number of ways to choose \( k \) items from \( n \) items.

In our exercise, we calculated \( \binom{9}{7} \) by simplifying it to \( \binom{9}{2} \). This simplification is a classic example of using combinatorics principles where \( \binom{n}{k} = \binom{n}{n-k} \), showcasing the symmetrical nature of combinations.

Permutations

Permutations refer to the number of ways to arrange a set of items where the order does matter.
In contrast to combinations, permutations focus on different sequences of arranging elements. This concept is important in:

• Understanding the difference between ordered and unordered selections.
• Solving problems that require arrangements rather than just selections.

To compute permutations, factorials are often used since they naturally account for all possible arrangements.
For example, with 3 unique items to arrange, the number of permutations is \( 3! \), which gives us 6 possible orders.
While the exercise focuses on combinations, knowing permutations helps grasp the bigger picture of arranging and selecting elements in various contexts.