Suppose that the format for license plates in a certain state is two letters followed by four numbers.
How many different plates can be made if repetitions of numbers and letters are allowed except that no plate can have four zeros?
To calculate the number of different license plates that can be made with the given format and restrictions, we can break down the problem into two parts: the letters and the numbers.
1. Calculating the number of possibilities for the letters:
Since repetitions of letters are allowed, and there are 26 letters in the English alphabet, we can have 26 choices for the first letter and 26 choices for the second letter. Therefore, the total number of possibilities for the two letters is:
26 * 26 = 676
2. Calculating the number of possibilities for the numbers:
Since repetitions of numbers are allowed, and there are 10 digits (0-9), we can have 10 choices for each of the four numbers. However, we need to exclude the possibility of having all four numbers as zeros (0000). Therefore, the total number of possibilities for the four numbers is:
10^4 - 1 = 10,000 - 1 = 9,999
3. Calculating the total number of different license plates:
To calculate the total number of different license plates, we multiply the number of possibilities for the letters by the number of possibilities for the numbers:
Total = Number of possibilities for letters * Number of possibilities for numbers
Total = 676 * 9,999
So, there can be a total of 6,754,324 different license plates that can be made if repetitions of numbers and letters are allowed, except that no plate can have four zeros.
To calculate the total number of different license plates that can be made in this state, we need to consider each position separately.
For the first position, we have two letters to choose from (A-Z) and no restriction.
For the second position, we again have two letters to choose from (A-Z) and no restriction.
For the third position, we have 10 numbers to choose from (0-9) and no restriction.
For the fourth position, we have 10 numbers to choose from (0-9) and no restriction.
For the fifth position, we have 10 numbers to choose from (0-9) and no restriction.
For the sixth position, we have 10 numbers to choose from (0-9) with the restriction that we cannot have four zeros.
Since we have four positions with letters and four positions with numbers, we multiply the number of choices for each position together to get the total number of different license plates:
2 * 2 * 10 * 10 * 10 * 9 = 36,000
Therefore, there can be a total of 36,000 different license plates that can be made in this state, considering the given conditions.
with no restriction at all : 26x26x10x10x10x10 = 6760000
but there are 26x26 plates that would end in 0000
so number of plates as required above = 6760000-676 = 6759324