Binomial Coefficients
This page is dedicated to providing tables and information regarding Binomial Coefficients. These coefficients are fundamental mathematical quantities that arise in various areas of mathematics, statistics, and computer science. They are commonly denoted by several notations, including $C(n, k)$, $nCk$, or perhaps most formally as $\binom{n}{k}$. These notations are all read as "n choose k".
At their core, binomial coefficients answer a very specific counting question within the field of combinatorics: Given a set of 'n' distinct items, how many different ways are there to choose a subset of exactly 'k' items from that set, where the order of selection does not matter? For example, if you have 5 distinct fruits and want to choose 2 of them, the binomial coefficient $\binom{5}{2}$ tells you the number of possible combinations. Choosing Apple then Banana is considered the same combination as choosing Banana then Apple.
The resource provided here serves to give you direct access to the values of these coefficients for a range of 'n' and 'k'. This might take the form of pre-calculated tables listing values for common combinations of 'n' and 'k'. Alternatively, it might reference structures like Pascal's Triangle, a triangular array where each number is the sum of the two numbers directly above it, and whose rows contain the binomial coefficients for increasing values of 'n'.
The mathematical formula used to calculate a binomial coefficient $\binom{n}{k}$ is derived from the principles of permutations and combinations. It is given by: $$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$ Here, the exclamation mark '$!$' denotes the factorial operation, where $m! = m \times (m-1) \times \dots \times 2 \times 1$. The formula effectively divides the number of ways to arrange 'n' items ($n!$) by the number of ways to arrange the 'k' chosen items ($k!$) and the number of ways to arrange the remaining $(n-k)$ items ($(n-k)!$), thereby removing the consideration of order within the chosen subset and the unchosen subset. Note that $0!$ is defined as 1. The coefficient is defined for non-negative integers $n$ and $k$, with $0 \le k \le n$.
Binomial coefficients are crucial in various areas:
- Probability Theory: They are fundamental in calculating probabilities in the Binomial Distribution, which models the number of successes in a fixed number of independent Bernoulli trials.
- Algebra: They appear as the coefficients in the expansion of powers of binomials, as described by the powerful Binomial Theorem: $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.
- Computer Science: Used in algorithms related to combinations, probability, and data structures.
- Statistics: Applied in sampling theory, design of experiments, and non-parametric statistics.
This resource provides a quick and convenient lookup for these important combinatorial numbers, saving the effort of calculating factorials and performing the division manually, especially for larger values of 'n' and 'k'.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| n = 1 | 1 | 1 | |||||||||
| 2 | 1 | 2 | 1 | ||||||||
| 3 | 1 | 3 | 3 | 1 | |||||||
| 4 | 1 | 4 | 6 | 4 | 1 | ||||||
| 5 | 1 | 5 | 10 | 10 | 5 | 1 | |||||
| 6 | 1 | 6 | 15 | 20 | 15 | 6 | 1 | ||||
| 7 | 1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | |||
| 8 | 1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | ||
| 9 | 1 | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | |
| 10 | 1 | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 |
| 11 | 1 | 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 |
| 12 | 1 | 12 | 66 | 220 | 495 | 792 | 924 | 792 | 495 | 220 | 66 |
| 13 | 1 | 13 | 78 | 286 | 715 | 1287 | 1716 | 1716 | 1287 | 715 | 286 |
| 14 | 1 | 14 | 91 | 364 | 1001 | 2002 | 3003 | 3432 | 3003 | 2002 | 1001 |
| 15 | 1 | 15 | 105 | 455 | 1365 | 3003 | 5005 | 6435 | 6435 | 5005 | 3003 |
| 16 | 1 | 16 | 120 | 560 | 1820 | 4368 | 8008 | 11440 | 12870 | 11440 | 8008 |
| 17 | 1 | 17 | 136 | 680 | 2380 | 6188 | 12376 | 19448 | 24310 | 24310 | 19448 |
| 18 | 1 | 18 | 153 | 816 | 3060 | 8568 | 18564 | 31824 | 43758 | 48620 | 43758 |
| 19 | 1 | 19 | 171 | 969 | 3876 | 11628 | 27132 | 50388 | 75582 | 92378 | 92378 |
| 20 | 1 | 20 | 190 | 1140 | 4845 | 15504 | 38760 | 77520 | 125970 | 167960 | 184756 |