Why is n choose k an integer

The initial elements of are 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 18, 19, 21, 22, 27, If is squarefree, then must belong to. It is very probable that is finite, but no proof is known. Now, squares larger than 4 and 9 might also divide , but by eliminating these two alone, the only possible for are 1, 2, 3, 4, 6, 9, 10, 12, 18, 33, 34, 36, 40, 64, 66, , , , , , , , , , , , , , , , and All of these but the last have been checked, establishing that there are no other such that is squarefree for.

Binomial coefficients are squares when is a triangular number , which occur for , 6, 35, , , , These values of have the corresponding values , 9, 50, , , , The binomial coefficients are called central binomial coefficients , where is the floor function , although the subset of coefficients is sometimes also given this name. Granville and Ramare proved that the only squarefree values are and 4.

Sander subsequently showed that are also never squarefree for sufficiently large as long as is not "too big. Most binomial coefficients with have a prime factor , and Lacampagne et al. Abramowitz, M. New York: Dover, pp. Comtet, L. Dordrecht, Netherlands: Reidel, Conway, J. In The Book of Numbers. New York: Springer-Verlag, pp. New York: Springer-Verlag, Feller, W. New York: Wiley, pp. Fowler, D. Monthly , , Graham, R.

Reading, MA: Addison-Wesley, pp. Granville, A. Binomial Coefficients Modulo Prime Powers. That's what I thought too, but he specified that he's not allowed to use anything but ints. That's pretty specific and doesn't allow Object types. Sign up or log in Sign up using Google.

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Search MathWorks. Open Mobile Search. Off-Canvas Navigation Menu Toggle. Main Content. Binomial Coefficients for Numeric and Symbolic Arguments Compute the binomial coefficients for these expressions. Handle Expressions Containing Binomial Coefficients Many functions, such as diff and expand , can handle expressions containing nchoosek.

Differentiate the binomial coefficient. Open Live Script. Input Arguments collapse all n — Number of possible choices nonnegative integer scalar. Example: 10 Example: int16 10 Data Types: single double int8 int16 int32 int64 uint8 uint16 uint32 uint Example: 3 Example: int16 3 Data Types: single double int8 int16 int32 int64 uint8 uint16 uint32 uint Set of all choices, specified as a vector.

Output Arguments collapse all b — Binomial coefficient nonnegative scalar value. C — All combinations of v matrix. Usage notes and limitations: When the first input, x , is a scalar, nchoosek returns a binomial coefficient. See Also perms. You have a modified version of this example. Do you want to open this example with your edits? No, overwrite the modified version Yes.