4. A square numeric keypad with a side of 4 has $\mathrm{i}$ keys numbered from 1 to 16. Agent 007 must press two keys simultaneously to infiltrate the enemy base, and if he makes a mistake, he will trigger the alarm. However, he only knows that the two keys are not adjacent (i.e., they do not share a side or a vertex). What is the probability that he will succeed in infiltrating?
(A) $\frac{1}{64}$
(B) $\frac{1}{78}$
(C) $\frac{1}{128}$
(D) $\frac{1}{156}$
(E) $\frac{1}{160}$.
4. The answer is (B). The required probability is the inverse of the number of possible pairs of non-adjacent keys. To count such pairs, we distinguish various cases:

(a) one of the keys is among the four central ones: then the other must be on the edge, and 5 adjacent keys must be excluded.

Possibilities: $4 \cdot 7=28$

(b) one of the keys is at a corner and the other is not (but not among the four central ones): only two adjacent edge keys are excluded.

Possibilities: $4 \cdot 6=24$

(c) both keys are at the corners: there are as many possibilities as there are pairs of corners.

Possibilities: $4 \cdot 3 / 2=6$

(d) both keys are on the edge but not at the corners: there are 8 keys, and each key excludes exactly two adjacent ones, leaving 5 valid ones (but each pair is counted twice).

Possibilities: $8 \cdot 5 / 2=20$

Thus, the total possibilities are 78.