Question

If $n$ positive integers are taken at random and multiplied together, then the probability that the last digit of the product is $2,4,6$or $8$, is

• A. $\frac{(4^n+2^n)}{5^n}$
• B. $\frac{\left(4^{n}x{2}^n\right)}{5^n}$
• C. $\frac{(4^n-2^n)}{5^n}$
• D. None of these

Answer 1

The correct option is C

$\frac{(4^n-2^n)}{5^n}$

Explanation for correct option

We know that,

Probability of an event is $=\frac{\mathrm{number}\mathrm{of}\mathrm{favorable}\mathrm{outcomes}}{\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$

Therefore the probability of integer ending in $1,2,3,4,6,7,8,9$ is $\frac{8}{10}=\frac{4}{5}$

Hence, the probability that the last digit of the product of $n$ integers is ${\left(\frac{4}{5}\right)}^n$

The probability that the last digit of the product of $n$ integers ends with $1,3,7,$ and $9$ is ${\left(\frac{2}{5}\right)}^n$

Hence, the required probability$={\left(\frac{4}{5}\right)}^n–{\left(\frac{2}{5}\right)}^n$

$=\frac{4^n–2^n}{5^n}$

Hence, option (C) is correct.