A state's license plates consist of four letters followed by three numerals, and 241 letter arrangements are not allowed. How many plates can the state issue?
isn't it:
[(26^4)+ (10^3)]-241
with no restrictions:
number of plates = 26^4 x 10^3
241 letter combinations are not allowed, there would be 241(10^3) or 241000 of those
number of plates = 26^4 x 1000 - 241x1000
=1000(26^4 - 241)
= 456,735,000
I am assuming that letters and numerals may be repeated.
To find the number of license plates the state can issue, we need to calculate the total number of possible arrangements.
1. Let's start by finding the total number of possible arrangements without any restrictions.
- Since there are four letters on the license plate, we have 26 choices of letters for each position.
- Similarly, there are ten possible choices (0-9) for each numeral position.
- Therefore, the total number of possible arrangements without restrictions is: 26^4 * 10^3.
2. However, we know that 241 letter arrangements are not allowed.
- Let's subtract these forbidden arrangements from the total number of possible arrangements.
- The total number of possible license plates will be: (26^4 * 10^3) - 241.
Now, let's calculate the answer:
Total number of plates = (26^4 * 10^3) - 241
Total number of plates ≈ 4,595,019,759
To find the number of license plates the state can issue, we need to calculate the total number of possible combinations and subtract the number of prohibited arrangements.
The possible combinations for the four letters can be calculated using the formula for permutations, since order matters:
Number of letter combinations = 26 (26 letters in the English alphabet) ^ 4 (four letters) = 26^4 = 456,976
Similarly, the possible combinations for the three numerals can be calculated using the formula for permutations:
Number of numeral combinations = 10 (10 numerals from 0 to 9) ^ 3 (three numerals) = 10^3 = 1,000
To get the total number of license plates, we multiply the number of letter combinations by the number of numeral combinations:
Total number of license plates = Number of letter combinations * Number of numeral combinations
Total number of license plates = 456,976 * 1,000 = 456,976,000
However, 241 letter arrangements are not allowed, meaning we subtract these prohibited arrangements from the total to get the final number of license plates the state can issue:
Final number of license plates = Total number of license plates - Number of prohibited arrangements
Final number of license plates = 456,976,000 - 241 = 456,975,759
Therefore, the state can issue 456,975,759 license plates.