C(n, k)

Definition

c(n, k), also written as 'n choose k', is a mathematical notation that represents the number of ways to choose k elements from a set of n distinct elements without regard for the order of selection. This concept is pivotal in combinatorics and is used extensively in probability and statistics, particularly in the expansion of binomial expressions as outlined in the binomial theorem.

5 Must Know Facts For Your Next Test

1. c(n, k) is calculated using the formula: $$c(n, k) = \frac{n!}{k!(n-k)!}$$, where n! denotes the factorial of n.
2. c(n, k) is symmetric, meaning that c(n, k) = c(n, n-k). This property reflects that choosing k items from n is equivalent to leaving out n-k items.
3. The value of c(n, k) is zero if k > n, since it's impossible to choose more elements than are available.
4. The sum of all combinations for a fixed n is given by the formula: $$\sum_{k=0}^{n} c(n, k) = 2^n$$, illustrating that there are 2^n possible subsets of a set with n elements.
5. In the context of the binomial theorem, c(n, k) represents the coefficients in the expansion of (a + b)^n, where each term's coefficient corresponds to how many ways you can choose terms from the expansion.

Review Questions

• How does the symmetry property of c(n, k) help in understanding combinations?
  • The symmetry property states that c(n, k) equals c(n, n-k). This means that the number of ways to choose k elements from n is exactly equal to the number of ways to leave out n-k elements. This property simplifies calculations and helps reinforce understanding that combinations can be viewed from different perspectives - either selecting items or omitting them.
• Discuss how c(n, k) relates to the coefficients in the binomial expansion and provide an example.
  • In the binomial expansion of (a + b)^n, each term's coefficient corresponds to a specific combination count represented by c(n, k). For example, in the expansion (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, the coefficients 1, 3, and 3 are derived from c(3, 0), c(3, 1), and c(3, 2) respectively. This shows how combinations directly influence the structure of polynomial expansions.
• Evaluate the impact of understanding c(n, k) on solving real-world problems involving probabilities and statistics.
  • Understanding c(n, k) is crucial for solving real-world problems because it provides a foundation for calculating probabilities in scenarios such as drawing cards from a deck or forming committees. By applying this concept, one can determine likelihoods associated with various outcomes. For instance, if you want to find the probability of drawing 2 red cards from a standard deck of 52 cards without replacement, you would calculate it using c(26, 2) for red cards and c(52, 2) for total draws. This illustrates how foundational knowledge in combinations leads to insights in practical applications like gambling odds or risk assessment.

Related terms

Binomial Coefficient: Another term for c(n, k), which counts the number of ways to choose k elements from a set of n elements.

Factorial: A mathematical function denoted by n!, representing the product of all positive integers up to n, used in calculating combinations.

Pascal's Triangle: A triangular array of numbers where each number is the sum of the two directly above it, with each row corresponding to the coefficients of the binomial expansion.