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

Hint: 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.