Out of all the 2-digit integers between 1 and 100, a 2-digit number has to be selected at random. What is the probability that the selected number is not divisible by 7?

1. Total number of 2-digit integers between 1 and 100: The 2-digit integers range from $10$ to $99$ inclusive.

   $$\text{Total outcomes, } N = 99 - 10 + 1 = 90$$

2. Number of 2-digit integers divisible by 7: The multiples of $7$ in the range $[10, 99]$ are:

   $$14, 21, 28, \dots, 98$$

   Since $14 = 7 \times 2$ and $98 = 7 \times 14$, the count is:

   $$n(\text{divisible by } 7) = 14 - 2 + 1 = 13$$

3. Number of 2-digit integers NOT divisible by 7:

   $$n(\text{not divisible by } 7) = 90 - 13 = 77$$

4. Probability:

   $$P = \frac{77}{90}$$

Hence, the correct option is 77/90 (Option D).