#### binomial (21.6.97)

One of my students tried to solve a very simple equation with Maple(release 4).

Maple gave the solution \(x=-3\) but by heart you can see it must be \(x=6\).

Can anybody explain me the structure of the binomialfunction in case of negative integers?

By the very deﬁnition of binomial coeﬃcients, this is a polynomial equation and you must bring it to
that form ﬁrst.

To ﬁnd out the deﬁnition of the binomial-function in Maple type this

and you will see that

This is simply the formular

binomial(n,k)=n*(n-1)*...*(n-k+1) / k*(k-1)*...*1
for n<0.

The help-function ?binomial does not tell the whole truth in this case.

A workaround would be to let fsolve only search for positive solutions:

The general deﬁnition (which works for any complex n, and nonnegative integer m) is
binomial(n,m) = n (n-1) ... (n-m+1)/m!

Thus binomial(-n,m) = (-1)^m binomial(n+m,m).

You might look at Graham, Knuth and Patashnik, ”Concrete Mathematics”, for more discussion and
applications of this.