Problem 9 - Algebra and Trigonometry

Question

Use the formula for \(_{n} C,\) to evaluate each expression. \(_{9} C_{5}\)

Step-by-Step Solution

Answer

The value of \(_{9}C_{5}\) is 126.

Step 1: Understand the Formula

The formula for combinations, denoted by \(_{n} C_{r}\) is given by \(_{n} C_{r} = \frac{n!}{r!(n-r)!}\) where `n` is the size of the set, `r` is the number of selections, and `!` denotes factorial, meaning the product of all positive integers up to that number.

Step 2: Input the Values

Following the formula for combinations, \(_{9}C_{5} = \frac{9!}{5!(9-5)!}\)

Step 3: Perform the Calculation

We calculate the factorial for 9, 5 and (9-5) respectively. Then, \(_{9}C_{5} = \frac{9!}{5!(9-5)!} = \frac{9*8*7*6*5!}{5!*4!} = \frac{9*8*7*6}{4*3*2*1} = 126\)

Key Concepts

Understanding Factorials

In mathematics, factorials are a way to describe the product of all integers from 1 up to a given number. It is denoted by an exclamation mark (!). For instance, 4 factorial is written as 4! and equals 4 × 3 × 2 × 1 = 24. The concept of factorials is integral in various areas of mathematics, especially in permutations and combinations.

Here are some important points to note about factorials:

• The factorial of 0 is defined as 1. This might seem counterintuitive at first, but it aligns with the properties of functions and combinatorics.
• Factorials grow rapidly. As the number increases, the factorial value grows very quickly. For example, 5! = 120, but 10! = 3,628,800.
• Factorials are often used in probability, statistics, and algebra to solve problems involving permutations and combinations.

Factorials provide the foundation for understanding combinations, as we use them when applying formulas like the binomial coefficient.

The Concept of Permutations

Permutations are all about arranging objects in specific orders. When using permutations, the order of selection matters. This is different from combinations, where the order does not matter. Permutations are calculated by using factorial notation.

If you have a set of `n` unique items and want to arrange them in a specific order, there are \[ n! = n \times (n-1) \times \ldots \times 1 \] ways to do so. However, if only a subset of `r` elements are selected from `n`, then the number of permutations is given by \[P(n, r) = \frac{n!}{(n-r)!}\]
When solving permutation problems:

• Always pay attention to whether order is important. If it isn't, then you're dealing with combinations, not permutations.
• Use permutations to solve problems like arranging people in a line or sequence of events where specific order is needed.

Permutations can often be confused with combinations, but the key distinction is the importance of order.

The Binomial Coefficient and Combinations

A binomial coefficient, represented as \(_{n}C_{r}\) or \( \binom{n}{r} \), indicates the number of ways to choose `r` elements from a set of `n` elements, without caring about the order of selection. The formula for the binomial coefficient is given by:

• \(_{n} C_{r} = \frac{n!}{r!(n-r)!}\)

This formula shows why understanding factorials is so crucial, as each term in the formula depends on calculating factorials.
To solve a combination problem, you need to:

• Identify the total number of elements in the set, \(n\).
• Determine the number of selections you need to make, \(r\).
• Apply the formula mentioned above to determine the number of combinations possible.

If you're asked to calculate \(_{9}C_{5}\), it means you're selecting 5 elements from a set of 9 without caring about the order. Using the formula, it computes to 126, showing exactly how many unique groups of 5 you can form from the 9 elements.