This is pretty straight forward. Writing "T" for a correct answer and "F" for an incorrect answer this is just asking how many way you can arrange "TTFF" and that is .

Similarly, "0 correct answers" is "FFFF" and there is only 1 way to do that. "1 correct answer " is "TFFF" and there are ways to arrange that (TFFF, FTFF, FFTF, FFFT). "2 correct answers" is 6 as above. "3 correct answers" if just "FTTT" which again gives 4, and "4 correct answers", "TTTT" can be done in just one way.c). If a candidate guesses at random where the four answers are to go and X is the number of correct guesses he makes, draw up the probability distribution for X in tabular form.

this question has been asked before in an old post, with no answer. Balls in bags, 4 answers to 4 exam questions.

For b), is it $ ^4C_2$? it is just a matter of choosing which 2 of the 4 are correct. for the remaining 2, there is only one way that they can be wrong, that is when the 2 are swapped. so,$ ^4C_2$. but i myself am not very persuaded by this approach. wondering if there is a better way which can be applied to other similar cases (because i don't think i can use this approach to do other similar cases, like finding the number of ways of getting 5 answers correct if there are 11 questions and 11 answers. btw it would be cool if somebody can tell me how to do this too)

For c) i spend so much time trying this, and i just cannot find the number of ways for 0 correct answer and 1 correct answer! this is not just the simple arranging objects that i have been doing, this involves arranging objects into fixed spaces. fyi the answer for 0 correct spaces is 9 ways and 1 correct answer is 8 ways. i googled and found this reply(for 0 correct answers):

"This is just the number of permutations of the set {1,2,3,4} that change eachelement. Each permutation that moves all elements can be represented in the form (1 x y z) or (1 x)(y z), where x = (1), z = (y), etc.

So the # of possible permutations that change all elements is just the number of possible configurations of (x y z) = plus the number of configurations of (1 x)(y z) which is 3, for a total of 9 ways."

but i do not understand the notation he is using. somebody please explain this