Suppose there's a company that's making ID badges

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

(A E I O U) -> choose 1

(0 1 2 3 4 5 6 7 8 9 ) -> choose 3

part 1

total number of badges

there are 4 spots in the ID --- --- --- ---

Case 1: first spot is a vowel and remaining are numbers

First spot can be chosen in 5 ways. The remaining 3 spots can be chosen in 10 ways each

Hence, 5 * 10 * 10 * 10

Case 2: second spot is a vowel and remaining are numbers

Second spot can be chosen in 5 ways. The remaining 3 spots can be chosen in 10 ways each

Hence, 5 * 10 * 10 * 10

Case 3 and 4: The same as above happens with the third and fourth spot being a vowel respectively.

Case 3: 5 * 10 * 10 * 10

Case 4: 5 * 10 * 10 * 10

Answer = 5 * 10 * 10 * 10 + 5 * 10 * 10 * 10 + 5 * 10 * 10 * 10 + 5 * 10 * 10 * 10

= 5 * 10 * 10 * 10 * 4

part 2

total number of badges that can be created without repeating a number

Case 1: first spot is a vowel and remaining are numbers

First spot can be chosen in 5 ways. The remaining 3 spots can be chosen in 10 ways, 9 ways and 8 ways

Hence, 5 * 10 * 9 * 8

Case 2: second spot is a vowel and remaining are numbers

Second spot can be chosen in 5 ways. The remaining 3 spots can be chosen in 10 ways, 9 ways and 8 ways

Hence, 5 * 10 * 9 * 8

Case 3 and 4: The same as above happens with the third and fourth spot being a vowel respectively.

Case 3: 5 * 10 * 9 * 8

Case 4: 5 * 10 * 9 * 8

Answer = 5 * 10 * 9 * 8 + 5 * 10 * 9 * 8 + 5 * 10 * 9 * 8+ 5 * 10 * 9 * 8

= 5 * 10 * 9 * 8 * 4