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? 0.00160 0.03846 0.19231 0.80769

Explanation:

Step1: Calculate total number of possible passwords

There are 26 letters in the alphabet. The number of ways to create a 5 - letter password with no repeated letters is given by the permutation formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 26\) and \(r=5\). So \(P(26,5)=\frac{26!}{(26 - 5)!}=26\times25\times24\times23\times22=7893600\)

Step2: Calculate number of passwords with no vowels

There are 5 vowels (a, e, i, o, u). So there are \(26-5 = 21\) non - vowels. The number of 5 - letter passwords with no repeated non - vowels is \(P(21,5)=\frac{21!}{(21 - 5)!}=21\times20\times19\times18\times17 = 2441880\)
The probability \(P(\text{no vowels})=\frac{21\times20\times19\times18\times17}{26\times25\times24\times23\times22}\approx0.309\) (not relevant for the second part)

Step3: Calculate probability that first letter is'm'

If the first letter is'm', then for the remaining 4 letters, we need to choose from the remaining \(26 - 1=25\) letters. The number of ways to choose the remaining 4 letters is \(P(25,4)=\frac{25!}{(25 - 4)!}=25\times24\times23\times22\)
The total number of 5 - letter non - repeating letter passwords is \(P(26,5)=26\times25\times24\times23\times22\)
The probability that the first letter is'm' is \(\frac{1\times25\times24\times23\times22}{26\times25\times24\times23\times22}=\frac{1}{26}\approx0.03846\)

Answer:

The approximate probability that the first letter of the password will be'm' is \(0.03846\)