C(n,r), also known as the combination formula or the binomial coefficient, is a fundamental concept in probability theory and combinatorics. It represents the number of ways to choose r items from a set of n items, without regard to order.
congrats on reading the definition of C(n,r). now let's actually learn it.
The formula for C(n,r) is: $C(n,r) = \frac{n!}{r!(n-r)!}$, where n! represents the factorial of n.
C(n,r) is used to calculate the number of possible combinations when selecting r items from a set of n items, without regard to order.
The value of C(n,r) is always an integer, as it represents the number of possible combinations.
C(n,r) is symmetrical, meaning that C(n,r) = C(n,n-r).
The sum of all C(n,r) values for a given n, where r ranges from 0 to n, is equal to 2^n, which represents the total number of possible subsets of a set with n elements.
Review Questions
Explain the purpose and application of the C(n,r) formula in probability topics.
The C(n,r) formula is used to calculate the number of possible combinations when selecting r items from a set of n items, without regard to order. This is a fundamental concept in probability theory, as it helps determine the number of favorable outcomes in a sample space. For example, when calculating the probability of drawing a specific combination of cards from a deck, the C(n,r) formula can be used to determine the total number of possible combinations, which is then used to calculate the probability of the desired outcome.
Describe the relationship between C(n,r) and the concept of permutations.
While C(n,r) represents the number of combinations, permutations deal with the arrangement of items in a specific order. The relationship between the two is that the number of permutations of r items from a set of n items is given by P(n,r) = n! / (n-r)!, whereas the number of combinations is given by C(n,r) = n! / (r!(n-r)!). The key difference is that permutations consider the order of the items, while combinations do not.
Analyze how the C(n,r) formula can be used to calculate the probability of specific events in probability topics.
The C(n,r) formula is a crucial tool in calculating probabilities in various probability topics. For example, when calculating the probability of obtaining a specific number of successes in a series of independent Bernoulli trials, the C(n,r) formula can be used to determine the number of ways the successes can occur, which is then multiplied by the probability of each success to obtain the overall probability. Additionally, in the context of discrete probability distributions, such as the binomial distribution, the C(n,r) formula is used to calculate the probability mass function, which is a fundamental concept in probability theory.
Related terms
Permutation: The arrangement of a set of items in a specific order, where the order of the items matters.
Factorial: The product of all positive integers less than or equal to a given positive integer.
Probability: The measure of the likelihood that an event will occur, expressed as a number between 0 and 1.