If x-1/x=3, what is x3-1/x3?
x^3 - (1/x)^3 = (x - 1/x) (x^2 + 1 + 1/x^2) = (x - 1/x) [(x - 1/x)^2 + 3] = 3 * (3^2 + 3) = 36.
How many solutions does the equation x1+x2+x3=11 have where x1, x2, and x3 are Natural Numbers?
We can solve this problem in a case where ``natural number" is meant to include zero (see Is zero a natural number? Why or why not?) Otherwise, we can just say that [math]x_1 = 1 + y_1[/math], etc., and solve the same problem with the sum on the right-hand side being smaller by 3.There is a general way to find the number of solutions to [math]x_1 + x_2 + ... + x_m = n[/math]. Suppose that we have [math]n[/math] balls and [math]m - 1[/math] dividers (all indistinguishable). Then there is a one-to-one mapping between solutions to the equation above and arrangements between balls and dividers (by taking [math]x_1[/math] to be the number of balls to the left of the first divider, [math]x_2[/math] the number of balls between the first and second, etc.) The number of such arrangements is just the number of ways to choose [math]n[/math] elements from a set of [math]n + m - 1[/math], which is [math]n + m - 1\choose m - 1[/math]. In this case, it would be [math]13 \choose 2[/math] or 78 (if we exclude zero, it would be [math]10 \choose 2[/math] or 45).