The binomial coefficient is an important measure in combinatorics based on factorials. The binomial coefficient calculates the number of combinations that result when a certain number of elements are drawn from a set, provided all elements are different. The binomial coefficient is expressed as n over k. The calculation is done by dividing the factorial of the total number of elements by the factorial of the number of drawn elements and the factorial of the number of remaining elements.
| ( | n | ) | = | n! |
| k | (k!(n-k)!) |
This is mathematically spoken the number of possibilities to create a subset of k elements from a set of n elements, or a combination without repetition with k draws from n elements.
Enter for n and k positive integers. n should be be larger as k. Then press Calculate to determine the binomial coefficient.
A well-known example of the application of the binomial coefficient is in Germany the lottery Lotto 6 aus 49. In this game, 6 out of 49 balls are drawn, each numbered consecutively, so they are clearly distinguishable. There are 49 over 6 different combinations, which amounts to almost 14 million. The bonus number is ignored here; it would simply multiply this number tenfold. The probability of getting six correct numbers with one field is 1 divided by 13983816. The probabilities for a smaller number of correct numbers are somewhat more complicated to calculate. For 5 correct numbers, the calculation formula using binomial coefficients is 6 over 5 times 43 over 1 for the number of desired possibilities. That is 258. In total, there are still 13983816 possibilities, so the chance of getting five correct numbers in the lottery is 258 to 13983816, or about 0.0018 percent.
The binomial coefficient is particularly large when, of course, n is very large and, on the other hand, k is close to n/2. If, however, k is very small or close to n, then the binomial coefficient is small. For k=n or k=0, it is 1; for k=n-1 or k=1, it is n. Furthermore, the binomial coefficient is the same for k=x and k=n-x. For k>n or k<0, it is 0, since there are no such possibilities.
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