Question

Form two digit numbers using the digit $0,1,2,3,4,5$ without repeating the digits.
$P$ is the event that the number so formed is even.
$Q$ is the event that the number so formed is divisible by $3$.
$R$ is the event that the number so formed is greater than $50$.

$S$ is the sample space.

Which of the following options is correct?

• A. $n\left(S\right)=25,n\left(P\right)=13,n\left(Q\right)=9,n\left(R\right)=5$
• B. $n\left(S\right)=25,n\left(P\right)=13,n\left(Q\right)=9,n\left(R\right)=2$
• C. $n\left(S\right)=25,n\left(P\right)=13,n\left(Q\right)=9,n\left(R\right)=4$
• D. $n\left(S\right)=25,n\left(P\right)=13,n\left(Q\right)=9,n\left(R\right)=8$

Answer 1

Answer: (C)

$n\left(S\right)=25,n\left(P\right)=13,n\left(Q\right)=9,n\left(R\right)=4$

The given digits are $0,1,2,3,4,5$
Number of two digit numbers that can be formed, as first place can be filed with 5 numbers and second place can be filed with another 5 numbers; $5\times 5=25$
Now, number is an even number, with 0 at the end there can be $5$ numbers, with $2$ at the end there can be $4$ numbers and with $4$ at the end there can be $4$ numbers. Hence, total numbers which are even $=4+4+5=13$
Numbers divisible by $3$ are, $12,15,21,24,30,42,45,51,54$
Thus, total numbers are, $9$
There are only $4$ number, viz. $51,52,53,54$ which are greater than $50$
Hence, $n\left(S\right)=25,n\left(P\right)=13,n\left(Q\right)=9,n\left(R\right)=4$