The Partial Derivative Puzzle

For a (smooth) function f(x,y) of two variables, we may take partial derivatives with respect to either variable. The first time we do this, we get the (first order) partial derivative functions f_x(x,y) and f_y(x,y), or, more briefly, simply

f_x, f_y.

If we differentiate one of these functions, we see three second-order partial derivatives,

f_xx, f_xy, f_yy.

And if we differentiate k times, then depending on how many times we differentiate with respect to x and how many times with respect to y, we get k+1 distinct partial derivative functions.

Now suppose that the number of independent variables is n. (For the above problem, n was 2). It is easy to see that the number of first order partial derivatives is n, but can you determine a formula for the number of partial derivatives of order k?

This puzzle courtesy of Rich Fabiano, via Fritz Keinert.

I give up, show me the solution.

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Last revised on 27 March 2002.