Problem 18 - Algebra 2

Question

Evaluate each expression. \(C(10,4)\)

Step-by-Step Solution

Answer

The value of \(C(10, 4)\) is 210.

Step 1: Understanding the Problem

We need to evaluate the combination formula, represented as \(C(n, r)\). The formula for combinations is \(C(n, r) = \frac{n!}{r!(n-r)!}\). In this problem, \(n = 10\) and \(r = 4\).

Step 2: Apply the Combination Formula

Substitute the values into the formula: \(C(10, 4) = \frac{10!}{4!(10-4)!}\). This will become \(\frac{10!}{4! \times 6!}\).

Step 3: Calculate Factorials

Compute the factorials: \(10! = 10 \times 9 \times 8 \times 7 \times 6!\). Therefore, \(C(10, 4) = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!}\).

Step 4: Simplify the Expression

Cancel out the \(6!\) in the numerator and denominator: \(\frac{10 \times 9 \times 8 \times 7}{4!}\).

Step 5: Compute 4!

Calculate \(4! = 4 \times 3 \times 2 \times 1 = 24\).

Step 6: Final Calculation

Now divide the result from the simplified expression: \(\frac{10 \times 9 \times 8 \times 7}{24}\). This simplifies to \(210\).

Step 7: Conclusion

Thus, the value of \(C(10, 4)\) is \(210\), indicating there are 210 ways to choose 4 items from a set of 10 without regard to order.

Key Concepts

Factorials

Factorials are a crucial mathematical concept often appearing in probability and combinatorics.
A factorial is the product of all positive integers up to a given number. It is denoted by the exclamation mark symbol. For example, the factorial of 4 is written as 4! and calculated as follows:

• 4! = 4 × 3 × 2 × 1 = 24

Factorials grow very rapidly with increasing numbers. For instance,

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

In simplifying combinations and permutations, canceling factorials can help reduce complex calculations.
In the combination problem, we canceled the common factorial component to simplify our work.

Permutations and Combinations

Permutations and combinations are methods to count possible arrangements or selections from a set of items.

• Permutations consider the order: choosing 3 books from a shelf of 5 where arrangement counts would be a permutation problem.
• Combinations disregard order: selecting 3 fruits from a basket of 5 without caring about order characterizes a combination.

Combinations are calculated using the specific formula:

• \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

In simpler words, the formula finds how many ways you can combine items without regard to order. For example, using our scenario, there are 210 ways to choose 4 items from a set of 10, regardless of the sequence of those choices.

Binomial Coefficient

The binomial coefficient is a key concept in both algebra and probability.
It’s deeply rooted in combinatorics and represents the number of ways to choose a subset of items. The binomial coefficient is denoted as \(C(n, r)\) or sometimes

• \(\binom{n}{r}\).

This notation represents the number of ways to select \(r\) items out of \(n\) total, and is calculated with the formula:

• \[ C(n, r) = \frac{n!}{r!(n-r)!} \]

This expression is derived from the broader binomial theorem in algebra, used to expand expressions of the form \((x + y)^n\).
In our example, the coefficient \(C(10, 4)\) equals 210, elucidating how to select 4 out of 10 items, a practical example of binomial coefficient application.