Question

A five-digit number is written down at random. The probability that the number is divisible by $5$ and no two consecutive digits are identical, is

• A. $\frac{1}{5}$
• B. $\frac{1}{5}.(\frac{9}{10}{)}^3$
• C. $(\frac{3}{5}{)}^4$
• D. none of these

Answer 1

Answer: (C)

$(\frac{3}{5}{)}^4$

The possible options for the last digits are 0 & 5 only. (2 out of 10)
Therefore number of possible options for second last digit are 9 out of 10 (cant be equal to last digit so 10-1)
Similarly,
Number of possible options for third last digit are 9 out of 10; fourth last digit are 9 out of 10.
For first digit there are 9-1=8 due to constraint out of 9 nos.
Therefore, probability that the number is divisible by 5 and no two consecutive digits are identical = $\frac{8}{9}\times \frac{9}{10}\times \frac{9}{10}\times \frac{9}{10}\times \frac{2}{10}={\left(\frac{3}{5}\right)}^4$