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Calculate the Central Binomial Coefficient

The central binomial coefficient is the binomial coefficient that is largest for an even-numbered upper value. The binomial coefficient for a given upper value is largest when the lower value is half the upper value, and since both values ​​must be natural numbers, this is only possible if the upper value is even. Therefore, the central binomial coefficient relates the upper value to the lower value. Note that the upper value here is 2n and the lower value is n, whereas the binomial coefficient has n at the top and k at the bottom. The actual calculation is the same.

( 2n ) = (2n)!
n (n!)2

For n enter a positive integer and press Calculate to determine the central binomial coefficient.

n:


( 2n ) =
n

This is:

The central binomial coefficient indicates the maximum number of possible subsets of a 2n-element set. The middle column in Pascal's triangle i the ascending sequence of central binomial coefficients, which are the central value of each odd row.

An example of the application of the central binomial coefficient is the following: an autonomous robot is supposed to move on a square grid and is only allowed to move right or up, one square at a time. It is supposed to find a path from the bottom left corner of the grid to the top right. If the grid is, for example, 10 × 10 squares in size, then there are many possible paths, each with 10 steps to the right and 10 steps up, for a total of 20 steps. The number of all different routes is determined by the combination of these steps, which is the central binomial coefficient. There is a total of 184,756 possible different, but equally long paths that the robot can take from the start to the finish.

Cases where the central binomial coefficient also occurs besides path planning, are the complexity of parallel processing, and random processes in financial mathematics.



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