Problem 28 - Algebra and Trigonometry

Question

List all the combinations of 5 objects \(a, b, c, d,\) and \(e\) taken 2 at a time. What is \(C(5,2) ?\)

Step-by-Step Solution

Answer

There are 10 combinations of the 5 objects taken 2 at a time. \( C(5,2) = 10 \).

Step 1: Understand the Problem

You need to list all combinations of 5 objects taken 2 at a time and also compute the combination notation value, which is noted as \( C(5,2) \).

Step 2: List All Combinations

List each pair of the objects \( a, b, c, d, e \).Combinations of 2 objects out of 5 are:\( (a, b), (a, c), (a, d), (a, e), (b, c), (b, d), (b, e), (c, d), (c, e), (d, e) \)

Step 3: Use Combination Formula

To find the number of combinations \( C(5,2) \), use the formula for combinations: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

Step 4: Substitute Values into the Formula

Substitute \( n = 5 \) and \( r = 2 \) into the formula: \[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} \]

Step 5: Calculate Factorials and Simplify

Calculate the factorials: \( 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120 \), \(2! = 2 \), and \(3! = 3 \cdot 2 \cdot 1 = 6 \).Then substitute back into the formula: \[ C(5, 2) = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10 \]

Step 6: State the Answer

Hence, \( C(5, 2) = 10 \). This confirms that there are 10 different combinations of the 5 objects taken 2 at a time.

Key Concepts

Combinatorial Mathematics

Combinatorial mathematics is a branch of mathematics that focuses on counting, arranging, and finding patterns in sets of elements. It helps in understanding how to count objects in a structured way to solve problems involving combinations and permutations. For instance, when selecting two objects out of a set of five, combinatorial mathematics helps us calculate the different combinations possible without listing them manually, which can be especially useful for larger sets. It is applied in a variety of fields like computer science, statistics, and optimization.

Factorials

Factorials are a fundamental concept in combinatorial mathematics. A factorial is the product of all positive integers up to a given number. It's denoted by an exclamation mark (e.g., 5!). For example, the factorial of 5 (denoted as 5!) is calculated as:
1. \(5 \times 4 \times 3 \times 2 \times 1 = 120\)
Factorials are crucial in the combination formula because they help in simplifying the calculations for selecting objects from a set. Remembering the basic factorial values and how to compute them will ease solving combination and permutation problems.

Combination Formula

The combination formula is used to count the number of ways to choose a subset of items from a larger set where order does not matter. This is often denoted as \( C(n, r) \) or \( \binom{n}{r} \). The formula is:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Here:
• \( n \) is the total number of items.
• \( r \) is the number of items to be chosen.
• \( n! \) is the factorial of \( n \).
• \( r! \) is the factorial of \( r \).
• \( (n-r)! \) is the factorial of \( n - r \).
By substituting the values into the formula, we can compute the combination efficiently. For example, \( C(5, 2) \) is calculated by:
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \times 3!} = \frac{120}{2 \times 6} = \frac{120}{12} = 10 \]

Permutations and Combinations

Permutations and combinations are two different ways of selecting items from a set. The key difference is that permutations consider the order of selection, whereas combinations do not.

• Permutations: Used when order matters.
  Formula: \[ P(n, r) = \frac{n!}{(n-r)!} \]
  Example: Arranging 3 out of 5 books on a shelf.


• Combinations: Used when order does not matter.
  Formula: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]
  Example: Selecting 2 out of 5 objects without regard to order.

Understanding the distinction and application of these concepts will help solve many real-life and mathematical problems efficiently.