Some information first:

In the general binomial model if we carry out n trials, with

probability p of success at each trial and probability q of failure,

where p+q = 1, then if we want a probability of say r successes, one

possible sequence is r successes followed by n-r failures.

The probability of this sequence is ppppp to r terms x qqqqq to n-r

terms.

However, we can get r successes in a whole range of sequences, the

number of sequences being the same as the number of possible sequences

of r p's and n-r q's.

n!

The number of sequences = -------- = C(n,r)

r! (n-r)!

So the total probability of r successes and n-r failures is

P(r) = C(n,r) p^r q^(n-r)

a)

P(2) = C(10,2) x (1/4)^2 x (3/4)^8 = 0.281

Considering no bionimal your chance would be

b)

The chance to answer 1 question correct is 25% (since 1/4 is correct and 3 are wrong)

Therefor on average 10 questions you will have 2.5 correct answers, ( considering that if 2 students, one will have 2 and one 3)

Ofcourse that's the case there is no penalty on wrong answer. I hope i was helpfull if you need anything contact me