Binomial Coefficient C(n, m)

The Binomial coefficient C(n, m) is defined for all n >= 0 and 0 <= m <= n. C(n, 0) = C(n, n) = 1 for n >= 0.

To compute C(n, m), n >= 0, 0 <= m <= n:
if m == 0 or m == n, C(n, m) = 1
if 2m > n, change m to m = n − m

Now C(n, m) = ((((((n * (n−1)) / 2) * (n−2)) / 3) ... ) * (n+1−m)) / m.
With this arrangement of alternating multiplies and divides, every divide gives a whole number result.

For example
C(9, 6) = C(9, 3) = (((9 * 8) / 2) * 7) / 3 = ((72 / 2) * 7) / 3 = (36 * 7) / 3 = 252 / 3 = 84.

n  C(n,0) C(n,1) C(n,2) C(n,3) C(n,4) C(n,5) C(n,6) C(n,7) C(n,8) C(n,9) C(n,10)
0       1
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
n  C(n,0) C(n,1) C(n,2) C(n,3) C(n,4) C(n,5) C(n,6) C(n,7) C(n,8) C(n,9) C(n,10)

And: Wolfram Function Evaluation -- Binomial

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This page accessed times since April 4, 2005.
Page created by: hjsmithh@sbcglobal.net
Changes last made on Monday, 30-Mar-09 09:15:19 PDT

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