Question

A two digit number is formed with digit $2,3,5,7,9$ without repetition, what is the probability that the number formed is
(i)An odd number
(ii)A multiple of $5?$

Answer 1

We use the permutation and combination formula to find the odd number and also find the multiple of the $5$. Because the combination of $2,3,5,7,9$ is used to make an odd number and multiple of the $5$. After the combination formula we use permutation.

Formula used: We use first combination formula and then permutation formula to calculate the required an odd number and the multiple of the $5$.
$P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$
Where $n\left( A \right) =$Number of the event
$n\left( S \right) =$Total number of the sample space

Complete step by step solution: Two digit number is formed with digit $2,3,5,7,9$ is \left\\{ {22,23,25,27,29,32,33,35,37,39,52,53,55,57,59,72,73,75,79,92,93,95,97,99} \right\\}$$$$ = Sample space = n\left( S \right)$$$$ = 25
Two digit odd number is formed with digit $2,3,5,7,9$ is \left\\{ {22,23,25,27,29,33,35,37,39,53,55,57,59,73,75,79,93,95,97,99} \right\\}$$$$ = Number of the event$= n\left( A \right) = 20$
Use the formula of the probability $P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$
Substitute the value of the number of the event and the sample space in the $P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$
$P\left( A \right) = \dfrac{{20}}{{25}}$
$20$ is divided by $25$ we get,
$P\left( A \right) = \dfrac{4}{5}$
Hence the probability of the odd number is $P\left( A \right) = \dfrac{4}{5}$
The number which are multiple of the $5$ are \left\\{ {25,35,55,75,95} \right\\}
Then the number of the event is $5$
Substitute the value of the number of the event and total number of the sample space in the $P\left( A \right) = \dfrac{{n\left( A \right)}}{{n\left( S \right)}}$
$P\left( A \right) = \dfrac{5}{{25}}$
$5$ is divided by the $25$ we get,
$P\left( A \right) = \dfrac{1}{5}$

Hence the probability that the number formed is a multiple of the $5$.

Additional Information: In mathematics, the method of arranging all the members of a set of data into some order is known as permutation. Permutation occurs when different ordering on certain finite sets. The combination is defined as a way of selecting items. In combination, unlike permutations, the order of selection does not matter.

Note: Student must have clear knowledge about permutation and combination . In questions, students must understand the digits $2,3,5,7,9$ to make the required number. They must have clear knowledge about making combinations and calculating total numbers possible.