Because 10 answers are known to be correct, only 10 answers are randomly chosen. The probability p of a correct answer is p=0.25. The probability of an incorrect answer is 0.75. The distribution is binomial because (0.25+0.75)^10=1.
The expansion is: p^10+10p(1-p)^9+10*9/2p^8(1-p)^2+10*9*8/3!p^7(1-p)^3+...
The coefficients are: 1 10 45 120 210 252 210 120 45 10 1, symmetrical. The probability of X>15 when 10 answers are known to be correct is in fact P(>5), which, because the distribution is discrete, is the sum of the individual probabilities from p^10 to 210p^6(1-p)^4.
This is: 0.000000954+0.000028610+0.000386238+0.003089905+0.0162220=0.019728 approx. (19.728%).