Questions: License plates in a particular state are to consist of 2 digits followed by 4 uppercase letters. a) How many different license plates can there be in this state if repetition of letters and numbers is permitted? b) How many different license plates can there be in this state if repetition of letters and numbers is not permitted? c) How many different license plates can there be in this state if the first and second digits must be odd, and repetition of letters and numbers is not permitted? d) How many different license plates can there be in this state if the first digit cannot be 4, and repetition of letters and numbers is not permitted? a) There are 45,697,600 different possible license plates if repetition of numbers and letters is permitted. b) There are 32,292,000 different possible license plates if repetition of numbers and letters is not permitted. c) There are 7,176,000 different possible license plates if the first and second digits must be odd, and repetition of letters and numbers is not permitted. d) There are different possible license plates if the first digit cannot be 4, and repetition of letters and numbers is not permitted.

Solution Steps

To find the total number of different license plates when repetition of letters and numbers is permitted, we calculate the choices for each position:

• For the 2 digits, there are \(10\) choices each (from \(0\) to \(9\)).
• For the 4 letters, there are \(26\) choices each (from \(A\) to \(Z\)).

Thus, the total combinations can be expressed as: \[ \text{Total combinations} = 10^2 \times 26^4 = 45697600 \]

When repetition of letters and numbers is not permitted, the calculation changes:

• For the 2 digits, there are \(10\) choices for the first digit and \(9\) choices for the second digit.
• For the 4 letters, there are \(26\) choices for the first letter, \(25\) for the second, \(24\) for the third, and \(23\) for the fourth.

Thus, the total combinations can be expressed as: \[ \text{Total combinations} = 10 \times 9 \times 26 \times 25 \times 24 \times 23 = 32292000 \]

For the scenario where the first and second digits must be odd and repetition is not permitted:

• The odd digits available are \(1, 3, 5, 7, 9\), giving us \(5\) choices for the first digit and \(4\) choices for the second digit.
• For the 4 letters, the choices remain the same as before: \(26\) for the first, \(25\) for the second, \(24\) for the third, and \(23\) for the fourth.

Thus, the total combinations can be expressed as: \[ \text{Total combinations} = 5 \times 4 \times 26 \times 25 \times 24 \times 23 = 7176000 \]

Final Answer