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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.

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 class=

Sagot :

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 [tex]n_1, n_2, \cdots, n_n[/tex] ways to be done, each thing independent of the other, the number of ways they can be done is:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

Item a:

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

Then, the possible number of passwords is:

[tex]N = 36^5 = 60466176[/tex]

Item b:

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

  • Starting with a letter, [tex]n_1 = n_3 = n_5 = 36, n_2 = n_4 = 10[/tex].
  • Starting with a digit, [tex]n_1 = n_3 = n_5 = 10, n_2 = n_4 = 36[/tex].

Then, the possible number of passwords is:

[tex]N = 36^3 \times 10^2 + 10^3 \times 36^2 = 5961600[/tex]

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

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