Suppose there's a company that's making ID badges for its employees

Question

Suppose there's a company that's making ID badges for its employees. IDs are made up of one vowel (A, E, I, O, U) and three numbers (0,1,2,3,4,5,6,7,8,9)

1.  what's the total number of badges that can be created?
2.  what's the total number of badges that can be created without repeating a number?

Answer 1

1- what's the total number of badges that can be created?

There are 5 options for the vowel and 10 options for each of the three numbers since there are 10 digits to choose from. Therefore, the total number of possible badges is:

5 options for vowel x 10 options for first digit x 10 options for second digit x 10 options for third digit = 5 x 10 x 10 x 10 = 5,000

So, 5,000 possible badges can be created.

2- what's the total number of badges that can be created without repeating a number?

For the first digit, there are 10 options. However, for the second digit, there are only 9 options left since one number has already been used. For the third digit, there are only 8 options left. Therefore, the total number of possible badges without repeating a number is:

5 options for vowel x 10 options for first digit x 9 options for second digit x 8 options for third digit = 5 x 10 x 9 x 8 = 3,600

So, there are 3,600 possible badges that can be created without repeating a number.