Combinatorics Question of finding number of ways for choosing correct answer In a multiple choice test there are six questions, each have four alternative answers of which only one answer is correct. If a candidate answers all the questions by choosing one answer for each question then number of ways to get exactly four correct answer is
There are $\binom{6}{4} = 15$ ways to choose $4$ question to answer right, $1^4$ ways to choose right answer for each of them and $3^2$ ways to answer two remaining questions. So there are $\binom{6}{4}\cdot 1^4\cdot 3^2 = 135$ ways to answer exactly $4$ questions right.
There are $\binom{6}{4} = 15$ ways to choose $4$ question to answer right, $1^4$ ways to choose right answer for each of them and $3^2$ ways to answer two remaining questions. So there are $\binom{6}{4}\cdot 1^4\cdot 3^2 = 135$ ways to answer exactly $4$ questions right.