Question

A password is to be made from a string of five characters chosen from the lowercase letters of the alphabet and the numbers 0 through 9.

Answer 1

Using the Fundamental Counting Theorem, it is found that for each case, the total number of outcomes is:

a) 60,466,176.

b) 5,961,600.

What is the Fundamental Counting Theorem?

It is a theorem that states that if there are n things, each with $n_1, n_2, \cdots, n_n$ ways to be done, each thing independent of the other, the number of ways they can be done is:

$N = n_1 \times n_2 \times \cdots \times n_n$

Item a:

No restrictions, hence for each of the five characters, there are 36 outcomes, hence $n_1 = \cdots = n_5 = 36$.

Then, the possible number of passwords is:

$N = 36^5 = 60466176$

Item b:

The letters and the digits have to be alternated, hence:

• Starting with a letter, $n_1 = n_3 = n_5 = 36, n_2 = n_4 = 10$.

• Starting with a digit, $n_1 = n_3 = n_5 = 10, n_2 = n_4 = 36$.

Then, the possible number of passwords is:

$N = 36^3 \times 10^2 + 10^3 \times 36^2 = 5961600$

To learn more about the Fundamental Counting Theorem, you can take a look at https://brainly.com/question/24314866