Number of different ways to answer a test with $25$ multiple-choice questions

If a multiple choice test has $25$ questions with $4$ alternatives per question, how many different ways can you answer the test? Is it $\;25 \times 4\;$ or $\;4^{25}\;$ and why? I don't understand....

Hint: Assume you had only $2$ questions with $4$ alternatives and ask yourself the question "in how many ways you can answer them". If you choose alternative $a$ in the firsts question you can choose alternative $a,b,c$ or $d$ in the second question, which gives you $4$ ways. Repeat for choice $b,c$ and $d$ in the first question, so you have in total $$4\times4=4^2$$ ways. Now, apply this result in the 25 questions.

Think about the simpler problem where there are only 2 questions with 4 alternatives per question. Are there $2\times 4 = 8$ or $4^2 = 16$ different ways to submit answers for this test?

The answer is $4^{25}$.

Think of it like this.

If you had $1$ question, you would have $4$ ways of answering.

If you had $2$ questions, you would have $4$ ways of answering question $1$. For each of the $4$ answers in question $1$, you would have $4$ ways of answering question $2$, so you have $4^2$ ways of answering two questions.

If you had three questions, you would have $4$ ways of answering question $1$, for each answer there would be $4$ ways of answering question $2$, for each answer for question $2$, there would be $4$ ways of answering question $3$. So you would have $4^3$ ways of answering three questions.

Increasing the number of questions by $1$ will lead to number of ways of answering being multiplied by $4$.

So for $25$ questions, you will have $4^{25}$ ways of answering.

If only one option out of four is to be selected as your answer to the multiple choice question then yes there are 4 ways to answer each question.

But if you are to select 1 or 2 or 3 or all 4 options as your answer then according to me there are 15 ways to answer a single multiple choice question with 4 options.

( 4 ways of selecting 1 option + 6 ways of selecting 2 options + 4 ways of selecting 3 options + 1 way of selecting all 4 options)

Hence for 25 questions it becomes 15 to the power 25 ways instead of 4 to the power 25.

• $16^{25}$ since you can choose none. Commented Oct 3, 2015 at 15:05

$4^{25}$. There are $4$ ways to choose an answer to each question. By fundamental counting principle, you multiply the number of choices at each step. We have $4\times4\times4\times\cdots= 4^{25}$ choices