Four questions are guessed randomly. Each question has 5 choices, one of which is correct.

So there are 625 permutations of answers and a 1/625 chance of getting every question right.

The easiest way to solve this is to use complementary counting. Find the number of cases where he gets 1 correct or none correct.

There is a 4/5 chance of getting each question WRONG. So the probability of getting them all wrong is 256/625.

The probability of getting ONE AND ONLY ONE correct is 1/5*4/5*4/5*4/5(the order doesn't matter here)=64/625.

Adding them up together, we get 320/625. But these are only the cases that we don't want, so we have to subtract from the total.

625-320=305. So the answer is 305/625=61/125.

I will leave you to check any possible errors. But you get the idea