How many different license plate numbers can be made using 4 letters followed by 3 digits, if
(a) Letters and digits may be repeated?Choose the 1st letter any of 26 ways.
Choose the 2nd letter any of 26 ways.
That's 26×26 ways to pick the first 2 characters.
Choose the 3rd letter any of 26 ways.
That's 26×26×26 ways to pick the first 3 characters.
Choose the 4th letter any of 26 ways.
That's 26×26×26×26 ways to pick the first 4 characters.
Choose the 1st digit, the 5th character, any of 10 ways.
That's 26×26×26×26×10 ways to pick the first 5 characters.
Choose the 2nd digit, the 6th character any of 10 ways.
That's 26×26×26×26×10×10 ways to pick the first 6 characters.
Choose the 3rd digit, the 7th character any of 10 ways.
That's 26×26×26×26×10×10×10 ways to pick the 7 characters.
Answer: 26×26×26×26×10×10×10 = 264103 = 456976000 ways.
(b) Letters may be repeated, but digits are not repeated?Choose the 1st letter any of 26 ways.
Choose the 2nd letter any of 26 ways.
That's 26×26 ways to pick the first 2 characters.
Choose the 3rd letter any of 26 ways.
That's 26×26×26 ways to pick the first 3 characters.
Choose the 4th letter any of 26 ways.
That's 26×26×26×26 ways to pick the first 4 characters.
Choose the 1st digit, the 5th character, any of 10 ways.
That's 26×26×26×26×10 ways to pick the first 5 characters.
Choose the 2nd digit, the 6th character any of the 9 remaining ways.
That's 26×26×26×26×10×10 ways to pick the first 6 characters.
Choose the 3rd digit, the 7th character any of the 8 remaining ways.
That's 26×26×26×26×10×10×10 ways to pick the 7 characters.
Answer: 26×26×26×26×10×9×8 = 264·P(10,3) = 329022720 ways
Edwin
