N!/(k!(n-k)!)

Definition

The expression $$\frac{n!}{k!(n-k)!}$$ represents the number of ways to choose k elements from a set of n elements without regard to the order of selection, commonly referred to as combinations. This concept is essential for understanding various combinatorial problems and is closely tied to the binomial theorem, which describes the expansion of binomial expressions and includes terms involving combinations.

5 Must Know Facts For Your Next Test

1. The formula $$\frac{n!}{k!(n-k)!}$$ is derived from counting permutations where order does not matter.
2. Combinations are used in probability and statistics to determine how likely certain outcomes are when selecting items.
3. The values of combinations can be found using Pascal's Triangle, where each entry corresponds to a binomial coefficient.
4. If k = 0 or k = n in the expression $$\frac{n!}{k!(n-k)!}$$, the result is always 1, as there's exactly one way to choose nothing or everything.
5. The symmetry property of combinations states that $$C(n, k) = C(n, n-k)$$, meaning choosing k elements is the same as choosing n-k elements.

Review Questions

• How does the expression $$\frac{n!}{k!(n-k)!}$$ relate to real-world scenarios such as lottery or team selection?
  • The expression $$\frac{n!}{k!(n-k)!}$$ is crucial in scenarios like lotteries or team selections where order does not matter. For instance, when picking a lottery ticket from a pool of numbers, we are interested in how many combinations can be formed without considering the order in which the numbers are drawn. This expression helps us calculate the total possible outcomes, allowing for better understanding and strategies in these situations.
• Discuss how combinations can be applied in probability theory and their importance in calculating outcomes.
  • Combinations play a significant role in probability theory by helping calculate the likelihood of different outcomes when selecting items from a larger set. When determining probabilities, we often need to know how many favorable outcomes exist compared to the total number of possible outcomes. By using $$\frac{n!}{k!(n-k)!}$$ to count these combinations, we can derive probabilities for events based on various selections and ensure accurate assessments in experiments or games of chance.
• Evaluate the significance of Pascal's Triangle in understanding combinations and their properties such as symmetry.
  • Pascal's Triangle is highly significant for understanding combinations because it visually represents binomial coefficients corresponding to each combination value calculated using $$\frac{n!}{k!(n-k)!}$$. Each row of the triangle corresponds to the values of n, while each position within that row gives us the value for different k. Additionally, it demonstrates properties such as symmetry, where each number mirrors its counterpart across the center line. This geometric representation not only aids in calculating combinations quickly but also enhances comprehension of their relationships within combinatorial mathematics.

Related terms

Factorial: The product of all positive integers up to a given number n, denoted as n!, which is foundational in calculating permutations and combinations.

Binomial Coefficient: Another term for the combination represented by $$\frac{n!}{k!(n-k)!}$$, often denoted as $$C(n, k)$$ or $$\binom{n}{k}$$.

Pascal's Triangle: A triangular array of numbers where each number is the sum of the two directly above it, which visually represents the coefficients of the binomial expansion and the values of combinations.