C(n, k)

Definition

c(n, k), also known as 'n choose k,' represents the number of ways to select a subset of k elements from a larger set of n elements without regard to the order of selection. This mathematical concept is essential in combinatorics and forms the basis for understanding combinations, where order does not matter, contrasting with permutations where it does. The formula for c(n, k) is given by $$c(n, k) = \frac{n!}{k!(n-k)!}$$.

5 Must Know Facts For Your Next Test

1. c(n, 0) = 1 for any n, as there is exactly one way to choose no items from a set.
2. c(n, n) = 1 for any n because there is only one way to choose all items from a set.
3. The value of c(n, k) is symmetric; that is, c(n, k) = c(n, n-k).
4. The sum of all combinations for a given n is equal to 2^n, reflecting the total number of subsets of a set.
5. c(n, k) can be computed recursively using the identity c(n, k) = c(n-1, k-1) + c(n-1, k).

Review Questions

• How does c(n, k) illustrate the difference between combinations and permutations?
  • c(n, k) highlights the key distinction between combinations and permutations by focusing on selections where order does not matter. In combinations, such as c(n, k), choosing items A and B is considered the same as choosing B and A; hence the arrangement isn't counted multiple times. In contrast, permutations account for different arrangements of those items and would give a different count depending on their order.
• Discuss the significance of the properties of c(n, k), particularly its symmetry and its relationship with factorials.
  • The symmetry property of c(n, k) indicates that choosing k elements from n is the same as choosing n-k elements from n; this can be represented mathematically as c(n, k) = c(n, n-k). This relationship simplifies calculations in combinatorial problems. Additionally, since c(n, k) involves factorials in its formula (specifically n!, k!, and (n-k)!), it emphasizes how quickly arrangements can grow as set sizes increase and connects to fundamental principles in probability and statistics.
• Evaluate how the concept of c(n, k) can be applied to real-world situations involving probabilities and decision-making processes.
  • c(n, k) has significant applications in real-world situations such as determining probabilities in games of chance or assessing decision-making scenarios. For instance, when evaluating the likelihood of drawing specific cards from a deck or selecting committee members from a larger group, c(n, k) helps quantify possible outcomes. Understanding these combinations allows individuals and organizations to make informed decisions based on potential scenarios and risks associated with different selections.

Related terms

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

Combination:

A selection of items from a larger set where the order of selection does not matter, represented mathematically as c(n, k).

Binomial Coefficient: Another term for c(n, k), representing the coefficients in the expansion of a binomial expression, which can be calculated using combinations.