binomial coefficients

Binomial coefficients, denoted by $n\choose k$ or $C(n,k)$

are so named because they appear as coefficient in the expansion of $(a+b)^n$

. In particular $C(n,k)$

is the coefficient of the terms $a^kb^{n-k}$ and $a^{n-k}b^k$ . It holds

${n\choose k} = \frac{n!}{k!(n-k)!}.$

Binomial coeffients arise in many mathematical problems, especially combinatoris. For example, there are $C(n, k)$ ways to choose $k$ elements from a set of $n$ objects.

The first nontrivial binomial coefficients (those for which $2 \le k \le n-2$ ) are 6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, 105, 120, 126, 136, 153, 165, 171, 190, 210, 220 more terms

Below, the spiral pattern of nontrivial binomial coefficients up to $256^2$ . See the page on prime numbers for an explanation and links to similar pictures.

The airy spirals that appear in the drawing are caused by the triangular numbers, which are a subset of binomial coefficients.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many binomial coefficients are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Binomial Coefficient
Wikipedia, Binomial coefficient
OEIS, Sequence A006987

Binomial coefficients can also be... (you may click on names or numbers and on + to get more values)

aban 10 15 20 + 999949000753 abundant 20 36 56 + 49995000 Achilles 29403 74024028 27665282700 + 1665678432600 admirable 20 56 66 + 523776 alternating 10 21 36 + 985036305 amenable 20 21 28 + 999961560 amicable 220 17296 122265 9363584 apocalyptic 220 286 528 + 29890 arithmetic 15 20 21 + 9997156 astonishing 15 Bell 15 betrothed 2024 brilliant 10 15 21 + 998977951 c.decagonal 1711 2467531 3558178261 5130890585101 c.heptagonal 253 64261 16322041 + 1053000152821 c.nonagonal 10 28 55 + 9999993493045 c.octagonal 1225 1413721 1631432881 1882672131025 c.pentagonal 276 1891 88831 + 2965643812176 c.triangular 10 136 1891 + 2677903145191 cake 15 378 canyon 105 325 406 + 98765012346 Carmichael 561 8911 10585 + 921323712961 congruent 15 20 21 + 9992685 constructible 10 15 20 + 2147516416 Cunningham 10 15 28 + 6294047334435 Curzon 21 78 105 + 199970001 cyclic 15 35 91 + 9992685 D-number 15 21 1953 + 5572791 d-powerful 153 378 1326 + 9792525 de Polignac 46971 79003 93961 + 99934453 decagonal 10 126 1540 + 2367195337045 deceptive 91 703 8911 + 99851537521 deficient 10 15 21 + 9992685 dig.balanced 10 15 21 + 199850028 double fact. 15 105 Duffinian 21 35 36 + 9956953 eban 36 56 66 + 40002040 economical 10 15 21 + 19961721 emirpimes 15 1891 3403 + 97531561 equidigital 10 15 21 + 19961721 eRAP 20 29938677951 412156185486 esthetic 10 21 45 + 5676765 Eulerian 66 120 evil 10 15 20 + 999916840 factorial 120 fibodiv 28 969 Fibonacci 21 55 Friedman 126 153 45760 + 857395 frugal 117855 118341 130816 + 957053125 gapful 105 120 165 + 99994143600 Gilda 78 220 330 + 990 happy 10 28 70 + 9988215 harmonic 28 496 8128 + 137438691328 Harshad 10 20 21 + 9999171820 heptagonal 55 286 27405 + 593128762435 hex 91 8911 873181 + 821574197731 hexagonal 15 28 45 + 9999997965180 highly composite 36 120 25200 + 7207200 hoax 84 136 364 + 99793128 Hogben 21 91 703 + 4734390674091 house 78 1716 7759830 hyperperfect 21 28 325 + 137438691328 iban 10 20 21 + 742371 idoneal 10 15 21 + 1365 inconsummate 84 276 630 + 980700 interprime 15 21 45 + 99906180 Jacobsthal 21 171 1365 Jordan-Polya 36 120 junction 105 210 406 + 99892045 Kaprekar 45 55 703 + 500000500000 katadrome 10 20 21 + 986310 Lehmer 15 91 435 + 999406030321 lonely 120 Lucas 5778 lucky 15 21 105 + 9988215 Lynch-Bell 15 36 126 + 416328 magic 15 2925 magnanimous 20 21 56 + 6824665 metadrome 15 28 35 + 2346 modest 406 666 3081 + 1984027528 Moran 21 45 84 + 666 Motzkin 21 mountain 120 153 165 + 1456893210 narcissistic 153 nialpdrome 10 20 21 + 222222111111 nonagonal 325 969 82621 + 1353857339341 nude 15 36 55 + 488234376 O'Halloran 20 36 84 924 oban 10 15 20 + 990 octagonal 21 560 5985 + 1086210502741 odious 21 28 35 + 999961560 palindromic 55 66 171 + 6874200024786 pancake 56 1771 panconsummate 10 15 20 + 231 pandigital 15 21 78 + 9654871320 partition 15 56 231 792 pentagonal 35 70 210 + 9995127883720 perfect 28 496 8128 + 137438691328 pernicious 10 20 21 + 9939111 Perrin 10 persistent 1520843976 2718609453 3574816290 + 97486512903 plaindrome 15 28 35 + 3555557777778 Poulet 561 2701 4371 + 9999899578441 power 36 1225 19600 + 1882672131025 powerful 36 1225 19600 + 1882672131025 practical 20 28 36 + 9965880 prim.abundant 20 56 66 + 95548245 primorial 210 productive 28 36 66 + 211876 pronic 20 56 210 + 323015470680 pseudoperfect 20 28 36 + 33550336 repdigit 55 66 666 repfigit 28 repunit 15 21 91 + 4734390674091 Rhonda 244856171115 Ruth-Aaron 15 78 105 + 921236838753 Saint-Exupery 780 497640 25743900 + 130459630080 Sastry 528 715 self 20 165 378 + 999246160 semiprime 10 15 21 + 99214741 sliding 20 70 1001 1330 Smith 378 666 861 + 99906180 sphenic 66 70 78 + 99934453 square 36 1225 19600 + 1882672131025 star 253 49141 9533161 + 358770992581 straight-line 210 630 666 741 strobogrammatic 1001 8008 super Niven 10 20 36 + 20000100000 super-d 105 190 231 + 9916831 superabundant 36 120 25200 + 7207200 tau 36 56 84 + 999246160 taxicab 61249825000 174882006936 tetrahedral 10 20 35 + 9999481661774 tetranacci 15 56 triangular 10 15 21 + 10000002437316 uban 10 15 20 + 8088059000035 Ulam 28 36 126 + 9956953 undulating 171 252 595 + 575757 unprimeable 325 1140 1330 + 9997156 untouchable 120 210 276 + 990528 upside-down 28 55 91 + 485628284526 vampire 14340690 1465813440 1563429820 + 7589181600 wasteful 20 28 36 + 9997156 weird 70 Wieferich 6161805 Zeisel 105 Zuckerman 15 36 816 + 911111328 Zumkeller 20 28 56 + 98790 zygodrome 55 66 666 + 5553333000111