What is the probability that x^2+7x+k is factorable if 0 is less than or equal to k is less than or equal to 20 and k is an integer?

The probability would be 1/7.

To factor trinomials in the form x²+bx+c, we look for factors of c that sum to be.  In our case, we want factors of k that sum to 7.  Since we know k must be an integer, we will list the ways to sum to 7 with positive integers:
1+6
2+5
3+4

This is where the 3 comes in the probability.

Since 0≤k≤20, there are 21 different integers that k could equal.  Therefore our probability is 3/21 = 1/7.