a state's license plate consist of 3 letters followed by 4 numerals, and 246 letter arrangements are not allowed. how many plates can be issued

(26^3-246)*10^4 = 173,300,000

26 letters

26 choices for first letter
26 choices for second letter assuming that repeats are allowed
and 26 choices for third letter
so
26^3 -246 for the letters
for the numbers
zero through 9 allowed, total of 10
again is repeats are allowed
10^4 or ten thousand
so in the end
(26^3 - 246)* 10,000

To calculate the number of plates that can be issued, we need to determine the total number of possible combinations and then subtract the number of prohibited arrangements.

Step 1: Calculate the number of possible combinations
Since the license plate consists of 3 letters followed by 4 numerals, we need to determine the possible combinations for each part.

For the letters:
There are 26 letters in the English alphabet, so the number of possible combinations for the letters is 26 * 26 * 26 = 17,576.

For the numerals:
There are 10 numerals (0-9), so the number of possible combinations for the numerals is 10 * 10 * 10 * 10 = 10,000.

Step 2: Subtract the prohibited arrangements
Given that there are 246 prohibited letter arrangements, we need to subtract this number from the total possible combinations.

Total number of plates = (Number of possible letter arrangements) * (Number of possible numeral combinations) - (Number of prohibited arrangements)
= 17,576 * 10,000 - 246
= 175,760,000 - 246
= 175,759,754

Therefore, the number of plates that can be issued is 175,759,754.

To determine the number of license plates that can be issued, we need to subtract the number of letter arrangements not allowed from the total number of possible arrangements.

First, let's calculate the total number of possible letter arrangements. There are 26 letters in the English alphabet, so for each of the three letters on the license plate, there are 26 options. Thus, the total number of possible letter arrangements is 26 x 26 x 26 = 17,576.

Next, we need to subtract the number of letter arrangements that are not allowed, which is given as 246.

Therefore, the number of plates that can be issued is 17,576 - 246 = 17,330.