C(n, k)

Definition

The term c(n, k), also known as 'n choose k' or binomial coefficient, represents the number of ways to choose k elements from a set of n distinct elements without regard to the order of selection. This concept is crucial in combinatorial mathematics and forms the foundation for understanding combinations and permutations, as well as being integral to the Binomial Theorem.

5 Must Know Facts For Your Next Test

1. The formula for c(n, k) is given by $$c(n, k) = \frac{n!}{k!(n-k)!}$$, where n! is the factorial of n.
2. c(n, k) is equal to c(n, n-k), meaning that choosing k elements from n is the same as leaving out (n-k) elements.
3. c(n, 0) equals 1 for any n because there is exactly one way to choose nothing from a set.
4. The values of c(n, k) can be found in Pascal's Triangle, where each entry is the sum of the two entries directly above it.
5. c(n, k) becomes significant in probability theory, as it helps calculate probabilities in binomial distributions.

Review Questions

• How does c(n, k) relate to permutations and why is this relationship important in combinatorial problems?
  • c(n, k) represents the number of combinations of choosing k items from n without considering order. In contrast, permutations take order into account and are calculated using c(n, k) alongside factorials. This relationship is crucial because many real-world problems require understanding both how many ways items can be chosen (combinations) and how they can be arranged (permutations), such as in scheduling or allocating resources.
• Demonstrate how to compute c(5, 3) and explain each step involved in using the formula.
  • To compute c(5, 3), we use the formula c(n, k) = $$\frac{n!}{k!(n-k)!}$$. Substituting n = 5 and k = 3 gives us c(5, 3) = $$\frac{5!}{3!(5-3)!}$$. Calculating this results in c(5, 3) = $$\frac{120}{6 \times 2}$$ = $$\frac{120}{12}$$ = 10. This shows there are 10 ways to choose 3 items from a set of 5.
• Evaluate the implications of c(n, k) in real-world scenarios like genetics or lottery systems and discuss how these implications affect outcomes.
  • In real-world scenarios such as genetics, c(n, k) helps determine probabilities of inheriting specific traits by calculating how many different combinations of genes can lead to certain phenotypes. In lottery systems, it determines the odds of winning by calculating how many ways a player can select winning numbers from a larger pool. Understanding these implications aids in making informed decisions based on likelihoods and enhances strategies for risk assessment and predictions in various fields.

Related terms

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

Permutation: An arrangement of items in a specific order; unlike combinations, permutations consider the order of selection.

Binomial Theorem:

A formula that provides a way to expand expressions that are raised to a power, using coefficients represented by c(n, k) which indicate the number of ways to choose terms.