Given, there are 30 questions in a multiple choice test. A student gets 1 mark each for an unattempted question, 4 marks for a correctly answered question, and 0 for a wrongly answered one. If the number of questions he does correctly be x and he scores 60, what are the number of possible values of x?
$$\begin{array}{c|c|c|} \text{Right (4 Marks)}& \text{Wrong (0 Marks)} & \text{Unattempted (1 Mark)} \\ \hline \text{15} & 15 & 0\\ \hline \text{14} & & 4\\ \hline \text{13} & & 8\\ \hline \text{12} & & 12\\ \hline \text{11} & & 16\\ \hline \text{10} & & 20\\ \hline \end{array}$$ Not filling up the "Wrong" column as there may be 1,2,3,...many possibilities for that column and it still wouldn't affect my answer. And I stop here as the maximum number of questions possible is 30. From here, I get my answer to be $6$, which is correct.
Now I was wondering if this could be solved using the multinomial theorem....by generating a function as follows:-
$$4\times x_1+0\times x^2 + 1\times x_3 = 60$$ $$4x_1+x_3 = 60$$
Is this the correct way to go about solving this question? If so, how would I find the number of solutions to the above equation??