Problem 18 - Algebra and Trigonometry

Question

Evaluate the expression. $$ C(9,2) $$

Step-by-Step Solution

Answer

\( C(9,2) = 36 \).

Step 1: Understanding the Expression

The expression \( C(9,2) \) represents a combination, which is used to determine the number of ways to choose 2 items from 9 items without regard to the order of selection. The general formula for a combination \( C(n, r) \) is \( \frac{n!}{r!(n-r)!} \).

Step 2: Identifying Variables

Identify that \( n = 9 \) and \( r = 2 \) in this problem. Substituting these into the formula, we have \( C(9,2) = \frac{9!}{2!\times(9-2)!} \).

Step 3: Calculating Factorials

Calculate the factorial values:- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)- \( 2! = 2 \times 1 = 2 \)- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)

Step 4: Simplifying the Expression

Substitute the factorials into the combination formula:\( C(9,2) = \frac{9 \times 8 \times 7!}{2 \times 1 \times 7!} \). Since \( 7! \) is in both the numerator and denominator, it cancels out, leaving:\( \frac{9 \times 8}{2 \times 1} \).

Step 5: Completing the Calculation

Perform the remaining arithmetic:\( \frac{72}{2} = 36 \). Thus, \( C(9,2) = 36 \).

Key Concepts

Factorials

Factorials are a mathematical operation that significantly influence how we calculate combinations and permutations. A factorial is represented by an exclamation mark (!) placed after a number. For any positive integer \( n \), the factorial is the product of all positive integers less than or equal to \( n \).
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials grow very quickly as the number increases, and they play a crucial role in counting problems.

• Zero Factorial: It's interesting to note that \( 0! \) is defined to be \( 1 \). This might seem strange, but it is essential for the integrity of mathematical formulas that involve factorials.
• Factorials in Combinations: In our problem, \( 9! \) and \( 2! \) are part of the formula for calculating combinations. Factorials help us determine the total number of possible arrangements of items and are essential in simplifying these expressions.

Combination Formula

The combination formula is a key tool in determining the number of ways to choose a subset of items from a larger set. In mathematics, combinations are expressed as \( C(n, r) \) or sometimes as \( \binom{n}{r} \). This notation represents the number of combinations possible when selecting \( r \) items from a total of \( n \) items, without considering the order.
The general formula for combinations is given by:\[C(n, r) = \frac{n!}{r!(n-r)!}\]

• Understanding the Formula: The formula involves dividing the total number of arrangements of \( n \) items, \( n! \), by the product of \( r!\) and \((n-r)! \). This adjustment ensures that permutations (where order matters) are not counted multiple times in the combination total.
• Application: In the example \( C(9,2) \), we calculated \( \frac{9!}{2! \, 7!} \). Factorials in the denominator help us adjust for the over-counting of arrangements within the subset and the remaining items.

Permutations

Permutations refer to the number of ways to arrange \( n \) items into a particular order. Unlike combinations, the order of items matters in permutations. If \( r \) objects are being chosen from a total set of \( n \), the number of permutations can be calculated using the formula:\[P(n, r) = \frac{n!}{(n-r)!}\]

• Compare with Combinations: Notice how permutations involve only \((n-r)!\) in the denominator compared to \( r!(n-r)! \) for combinations. This is because permutations take all possible orders into account.
• Example: If we had to find the number of ways to arrange 2 items from 9, we would calculate \( P(9, 2) = \frac{9!}{7!} \).

Using permutations and combinations effectively requires understanding when order is relevant to your problem scenario. This distinction is why factorials and their associated formulas are so powerful.