Question

Let $n$ positive integers taken randomly & multiplied together.Let $A_1$, $A_2$, $A_3$, $A_4$,$A_5$, $A_6$, $A_7$, be the events define as
$A_1$ = last digit of the product is not the number $1,3,7$ or $9$
$A_2$ = last digit of the product is among the number $1,3,5,7$ or $9$,
$A_3$ = last digit of the product is among the number $1,3,7$ or $9$
$A_4$ = last digit of lhe product 'is among the number $2,4,6$ or $8$.
$A_5$ = last digit of the product is the number $5$.
$A_6$ = last digit of the product is the number $0$
$A_7$ = last digit of the product is $1,2,3,4,6,7,8$ or $9$.On the basis of above information answer the following questions.

What is the probability that last digit of the product is $2,4,6$, or $8$ ?

• A. $\frac{6^n-4^n}{{10}^n}$
• B. $\frac{4^n-3^n}{5^n}$
• C. $\frac{4^n-2^n}{5^n}$
• D. $\frac{4^n-2^n}{{10}^n}$

Answer 1

The correct option is C $\frac{4^n-2^n}{5^n}$

$A_4=$ last digit in the product $x=a_1,a_2.....a_a$ is among the numbers $2,4,6$ or $8$, then none of the number end in $0$ or $5$ and one of the last digit must be even. Now out of $10$ numbers in which anyone of $n$ numbers may end, $8$ are favourable to the event that none of them end with $0$ or $5$. Hence $8^n$ is the number of ways in which we can exclude or $5$ & of these $4^n$ are the cases in which the last digit can be selected solely from the digit $1,3,7$ or $9$ Thus the favourable number of cases to the event that last digit in the product is $2,4,6$ or $8$ is $8^n-4^n$
$\therefore n\left(A_4\right)=8^n-4^n$
$\therefore P\left(A_4\right)=\frac{n\left(A_4\right)}{n\left(S\right)}=\frac{8^n-4^n}{{10}^n}=\frac{4^n-2^n}{5^n}$ Choice (c) is correct.
Note: The number of favourable cases of $A_4$ also be counted in the following way
$n\left(A_4\right)=n\left(S\right)-$ number of favourable cases of $A_2-$ of favourable cases of of $A_1$ ......(*) where number of favourable case of $A_7$ given by
$n\left(A_1\right)=n\left(S\right)-$ number of cases in which last digit in x is 5-number of ways in which 0 & 5 ~an be exclude $n\left(S\right)-n\left(A_5\right)-n\left(A_8\right)={10}^n-(5^n-4^n)-8^n={10}^n-8^n-5^n+4^n$
$\therefore n\left(A_4\right)={10}^n-({10}^n-8^n-5^n+4^n)=8^n-4^n$ by using (*)