Question

Let $A$ be the set of all positive prime numbers less than 30. Then find the number of different rational numbers whose numerator and denominator belongs to$A$.

Answer 1

Hint: This question can be solved using the concept of combinations. Combinations in mathematics can be simply explained as a mathematical technique that determines the number of arrangements possible for collection of items where the order of the selection does not matter. So in combinations we can select an item in any order. So by using this concept we can solve the above given question.

Complete step-by-step solution:
Given
$A:\;{\text{Set of all positive prime numbers less than 30}}...............\left( i \right)$
Now we have to find the number of different rational numbers whose numerator and denominator belongs to$A$.
So first let us list all positive prime numbers less than 30. It can be written as:
$A = 2,3,5,7,11,13,17,19,23,29$
So in total there would be 10 such numbers in$A$.
Now we know that the basic definition of a rational number is:
A rational number is a number that can be in the form $\dfrac{p}{q}$ where $p$ and $q$are integers and $q$ is not equal to zero.
So using the theory of combinations we can write:
Number of ways$p$can be selected from$A$: $^{10}{C_1} = \dfrac{{10!}}{{1!\left( {10 - 1} \right)!}} = 10......................\left( {ii} \right)$
Number of ways$q$can be selected from$A$:$^{\left( {10 - 1} \right)}{C_1}{ = ^9}{C_1} = \dfrac{{9!}}{{1!\left( {9 - 1} \right)!}} = 9...................\left( {iii} \right)$
So from (ii) and (iii) the number of different rational numbers whose numerator and denominator belongs to$A$can be written as:
$10 \times 9 = 90.................\left( {iv} \right)$

Therefore our final answer would be: $90$

Note: General formula to find combination in the form$^n{C_r}$is:
 $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
While doing a question involving combinations one should be careful about the calculation processes since it can be tricky as well as lengthy.
Also the above given method is recommended to solve similar questions involving combinations.