Problem 3 - Algebra 2

Question

Evaluate each expression. \(C(4,2)\)

Step-by-Step Solution

Answer

The value of \(C(4,2)\) is 6.

Step 1: Understand the Combination Formula

The expression \(C(n, r)\) represents the combination formula, which calculates the number of ways to choose \(r\) items from \(n\) items without regard to order. The formula is given by \[C(n, r) = \frac{n!}{r!(n-r)!}\].

Step 2: Plug in the Values

Given the expression \(C(4,2)\), we need to plug \(n = 4\) and \(r = 2\) into the combination formula. Thus, we have: \[C(4, 2) = \frac{4!}{2!(4-2)!}\].

Step 3: Calculate the Factorials

Calculate the factorials: \(4!\), \(2!\), and \((4-2)! = 2!\). \[4! = 4 \times 3 \times 2 \times 1 = 24\] \[2! = 2 \times 1 = 2\], thus \[(4-2)! = 2! = 2\].

Step 4: Substitute Factorials into the Formula

Substitute the calculated factorial values into the formula: \[C(4, 2) = \frac{24}{2 \times 2}\].

Step 5: Simplify the Expression

Simplify the expression: \[\frac{24}{4} = 6\]. Therefore, \(C(4, 2) = 6\).

Key Concepts

Factorials

Factorials are a fundamental part of combinatorics and play a crucial role in calculating combinations and permutations. A factorial, denoted by the symbol !, represents the product of a positive integer and all the integers below it. For example, the factorial of 4, written as 4!, is calculated as follows:

• Start with the number itself: 4
• Multiply it by one less than itself: 4 x 3 = 12
• Continue multiplying by decreasing integers down to 1: 12 x 2 = 24
• Finally multiply by 1 (though it doesn't change the value): 24 x 1 = 24

Therefore, 4! = 24. Factorials grow very quickly and are helpful in various mathematical computations, particularly in arrangements and selections.

Combination Formula

The combination formula is a method used to determine how many ways a certain number of items can be selected from a larger set when the order does not matter. The formula is represented as:\[C(n, r) = \frac{n!}{r!(n-r)!}\]where n is the total number of items to choose from, and r is the number of items to be chosen. Combinations differ from permutations in that order does not matter in combinations.For example, when using the formula to find C(4, 2), you are determining how many ways you can choose 2 items from a set of 4. Plugging in 4 for n and 2 for r gives us:\[C(4, 2) = \frac{4!}{2!(4-2)!}\]This simplifies to find the number of combinations, which we determined in the solution to be 6.

Mathematical Expressions

When you encounter mathematical expressions in combinatorics, it's important to understand the notation used as well as how each part relates to the whole equation. An expression like C(4, 2) denotes a specific operation as dictated by the combination formula. In the solved problem, the expression denotes selecting 2 items from a total of 4. Understanding each component of an expression is key:

• n is the total number of items available, here it is 4
• r is the number of items we want to choose, here it is 2
• Each factorial and subsequent operation must be carried out carefully for the accurate result

Breaking down these expressions into smaller, manageable parts allows you to solve them piece by piece.

Problem Solving Steps

In tackling combinatorics problems like evaluating C(4, 2), following structured problem-solving steps is pivotal. Here's a simplified guide drawing on the exercise:

• Start by understanding the problem: Identify whether it’s a permutation or a combination
• Connect the problem to the formula: For combinations, use the combination formula
• Write down the formula, substituting it with known values. Replace n and r with numbers if given directly
• Calculate any necessary factorials. Write down each multiplication step to avoid mistakes
• Substitute these values back into your formula and simplify. Work through division steps to simplify correctly

Each step, from understanding the type of problem to manually calculating factorial values, ensures clarity and accuracy while solving combinatorics expressions. This structure guarantees you'll get to the right answer methodically.