Combinations

Definition

Combinations are selections of items where the order does not matter. They are calculated using the binomial coefficient formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

5 Must Know Facts For Your Next Test

1. Combinations are used when the order of items does not matter, unlike permutations.
2. The formula for combinations is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ where $n$ is the total number of items and $k$ is the number of items to choose.
3. Combinations can be related to Pascal's Triangle, where each entry represents a combination value.
4. The concept of combinations is fundamental in probability theory for calculating possible outcomes.
5. For large values of $n$ and $k$, combinations can be approximated using Stirling's approximation for factorials.

Review Questions

• What is the difference between permutations and combinations?
• How do you calculate the number of combinations when choosing 3 items from a set of 10?
• Explain how Pascal's Triangle relates to combinations.

Related terms

Permutations: Arrangements of items where the order matters, calculated as $P(n, k) = \frac{n!}{(n-k)!}$.

Binomial Theorem:

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{(n-k)} b^k$$ describes the expansion of powers of binomials.

Factorial: $$ n! $$ (read as 'n factorial') is the product of all positive integers up to $n$, used in calculating permutations and combinations.