Question

You have 5 question multiple choice test. Each question has for choices. What the experimental probability that you will guess exactly three out of five questions correctly.

Answer 1

\[\dbinom{5}{3}(\frac{1}{4})^3(\frac{3}{4})^2\]

Answer 2

means three right, two wrong, and there are
\[\dbinom{5}{3}=\frac{5\times 4}{2}=10\] different ways to get this

Answer 3

There are 4 ways of answering each question,
Total no. of ways of answering the 5 questions= \(4\times 4\times 4 \times 4\times 4=4^5\)
There is only one way of answering each question correctly. Select 3 out of 5 questions= \[5C3= \frac{5!}{3!2!}=10\]
Now answer 3 questions with one way each.
And remaining two can be answered in 3*3=9 ways
Probability=\[\frac{10\times 9}{4^5}\]