a randomly generated password contains four characters. each of the four characters is either a lowercase…

Explanation:

Step1: Calculate total number of possible passwords

There are 26 lowercase letters and 10 digits (0 - 9), so a total of 36 characters. The number of ways to choose 4 non - repeating characters from 36 is given by the permutation formula $P(n,r)=\frac{n!}{(n - r)!}$, where $n = 36$ and $r=4$. So, $P(36,4)=\frac{36!}{(36 - 4)!}=\frac{36!}{32!}=36\times35\times34\times33=1413720$.

Step2: Calculate number of passwords with exactly one number

First, choose the position for the number in 4 ways. Then choose a number in 10 ways. Next, choose 3 non - repeating lowercase letters from 26 letters. The number of ways to choose 3 non - repeating lowercase letters from 26 is $P(26,3)=\frac{26!}{(26 - 3)!}=\frac{26!}{23!}=26\times25\times24$. The total number of passwords with exactly one number is $4\times10\times26\times25\times24 = 624000$.

Step3: Calculate the probability

The probability $P$ is the number of favorable outcomes divided by the number of total outcomes. So, $P=\frac{624000}{1413720}\approx0.4427\approx44%$.

Answer:

44%