Question

A student must answer $3$ out of $5$ essay questions on a test. In how many different ways can the student select the question?

Answer 1

To answer these types of questions, one must have basic knowledge about permutation and combination. Permutation can be defined or described as a way of arranging things or objects in a definitive order, on the other hand, the combination can be described as a way of selecting or choosing things or objects from a group of objects in a way that the order in which the objects are selected does not matter.

Complete step by step answer:
It is given that there are $5$ essay questions and a student has to select $3$ questions out of them. In this case, the order of the questions selected does not matter, therefore, we just need combinations of questions so that they are different questions.
Combinations can be calculated using the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
Where $n$is the number of objects or elements and $r$ is the number of objects that are chosen at a time.
In the question, the total number of questions given are $5$ , so $n=5$ and a student has to choose any $3$ questions out of them, therefore, $r=3$ .
Now, substitute the values in the above equation, to get,
$\Rightarrow {}^{5}{{C}_{3}}=\dfrac{5!}{3!\left( 5-3 \right)!}$
Also, we know that factorial of a number can be given as $n!=n\times \left( n-1 \right)\times \left( n-2 \right)......\left( n-n+1 \right)$
Simplifying the above equation by taking factorials of the number we get,
$\Rightarrow \dfrac{5!}{3!\left( 5-3 \right)!}=\dfrac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1\times 2\times 1}$
Now, further simplifying the above equation by cancelling the like terms and multiplying the remaining terms, we get,
$\Rightarrow \dfrac{5\times 2}{1\times 1}=10$

Therefore, there are $10\;$ ways of choosing $3$ essay questions out of the $5$ given questions.

Note: While solving such questions, students must be familiar with the factorial notations. Also, one should carefully think and then decide according to the data given in the question, whether combination or permutation has to be used while solving a given question.