The number of real solutions of the equation \((x^2 - 15x + 55)^{x^2-5x+6} = 1\) is:

The equation has 6 real solutions. A zero exponent gives $x=2,3$, while a base of 1 gives $x=6,9$. A base of $-1$ gives $x=7,8$, where the exponents are the even integers 20 and 30, respectively. Counting 4 solutions omits these last two cases.