Question

a website randomly creates an initial password for people when they first sign up for an account. the password consists of five letters, and cannot include numbers or special characters. the letters of the password cannot repeat. what is the approximate probability that a password will have no vowels? what is the approximate probability that the first letter of the password will be m?

Explanation:

Step1: Calculate total number of possible passwords

The total number of ways to form a 5 - letter non - repeating password from 26 letters is given by the permutation formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 26\) and \(r=5\). So \(P(26,5)=\frac{26!}{(26 - 5)!}=\frac{26!}{21!}=26\times25\times24\times23\times22=7893600\).

Step2: Calculate number of passwords with no vowels

There are 21 non - vowels. The number of 5 - letter non - repeating passwords with no vowels is \(P(21,5)=\frac{21!}{(21 - 5)!}=\frac{21!}{16!}=21\times20\times19\times18\times17 = 2441880\). The probability \(P_1\) that a password has no vowels is \(\frac{P(21,5)}{P(26,5)}=\frac{2441880}{7893600}\approx0.31\).

Step3: Calculate probability that first letter is 'm'

If the first letter is 'm', then we need to choose the remaining 4 non - repeating letters from the remaining 25 letters. The number of ways to do this is \(P(25,4)=\frac{25!}{(25 - 4)!}=\frac{25!}{21!}=25\times24\times23\times22 = 303600\). The probability \(P_2\) that the first letter is 'm' is \(\frac{P(25,4)}{P(26,5)}=\frac{303600}{7893600}\approx0.039\).

Answer:

The probability that a password will have no vowels is approximately \(0.31\).
The probability that the first letter of the password will be 'm' is approximately \(0.039\).