Problem 30 - Algebra and Trigonometry

Question

List all the combinations of 6 objects \(1,2,3,4,5,\) and 6 taken 3 at a time. What is \(C(6,3) ?\)

Step-by-Step Solution

Answer

The number of combinations is 20, and they are: \{1, 2, 3\}, \{1, 2, 4\}, \{1, 2, 5\}, \{1, 2, 6\}, \{1, 3, 4\}, \{1, 3, 5\}, \{1, 3, 6\}, \{1, 4, 5\}, \{1, 4, 6\}, \{1, 5, 6\}, \{2, 3, 4\}, \{2, 3, 5\}, \{2, 3, 6\}, \{2, 4, 5\}, \{2, 4, 6\}, \{2, 5, 6\}, \{3, 4, 5\}, \{3, 4, 6\}, \{3, 5, 6\}, \{4, 5, 6\}.

Step 1 - Understand the Problem

We need to find all the combinations of 6 objects \(1, 2, 3, 4, 5, 6\) taken 3 at a time.

Step 2 - Know the Combination Formula

Use the combination formula, which is \(C(n, k) = \frac{n!}{k!(n-k)!}\). Here, \(n = 6\) and \(k = 3\).

Step 3 - Calculate the Combinations

Calculate \(C(6, 3) \) using the formula: \( \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{6 \cdot 5 \cdot 4 \cdot 3!}{3! \cdot 3!} = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2 \cdot 1} = 20\). So, there are 20 combinations.

Step 4 - List All Combinations

List all the 3-object combinations from \(1, 2, 3, 4, 5, 6\): \{1, 2, 3\}, \{1, 2, 4\}, \{1, 2, 5\}, \{1, 2, 6\}, \{1, 3, 4\}, \{1, 3, 5\}, \{1, 3, 6\}, \{1, 4, 5\}, \{1, 4, 6\}, \{1, 5, 6\}, \{2, 3, 4\}, \{2, 3, 5\}, \{2, 3, 6\}, \{2, 4, 5\}, \{2, 4, 6\}, \{2, 5, 6\}, \{3, 4, 5\}, \{3, 4, 6\}, \{3, 5, 6\}, \{4, 5, 6\}.

Key Concepts

Combination Formula

Combinations in mathematics refer to the different ways you can select items from a larger set, where the order of selection does not matter. The combination formula you need to use is given by \( C(n, k) = \frac{n!}{k!(n-k)!} \). This formula is used to determine the number of ways to choose \( k \) items from \( n \) items without caring about the order.
For example, if you want to choose 3 out of 6 objects (like in our problem), you will use: \( C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} \).
This formula works by dividing the total number of possible selections (\( n! \)) by the number of selections in each subgroup (\( k! \)) and the number of ways to form groups not chosen (\( (n-k)! \)). This makes it versatile and easy to use for different numbers of objects.

Factorials

A factorial, denoted by an exclamation mark (\( ! \)), is a mathematical operation that multiplies a number by all positive integers less than itself.
For instance, \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \). You can think of it as a way to count permutations or ordered arrangements of objects.
In our case, we calculated:

• \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
B

Factorials grow very fast with the number of objects, which can sometimes make calculations tricky without a calculator. However, they play a crucial role in the combination formula, simplifying the counting process of various selections.

Binomial Coefficients

Binomial coefficients are numbers that arise in the expansion of a binomial raised to a power, represented as \( \binom{n}{k} = C(n, k) \). These coefficients are deeply related to combinations and appear in the binomial theorem for expanding expressions of the form \( (a+b)^n \).
Binomial coefficients tell you how many ways you can pick \( k \)-elements subsets from an \( n \)-element set. Using our example, \( \binom{6}{3} = C(6, 3) = 20 \), means there are 20 ways to choose 3 items from 6 items without considering the order.
This concept simplifies the process of solving problems from algebra to probability, making it an essential topic in mathematics.
In summary, learning about binomial coefficients helps with understanding the more extensive applications of combinations, including their use in probability and algebraic expansions.