Final answer:
Combinatorics, a branch of mathematics, is used to solve the problem of calculating how many unique license plates can be created with 6 non-repeating alphabet letters. Using permutations without repetition, it is found that 165,765,600 unique license plates are possible.
Step-by-step explanation:
The subject of this question is combinatorics, a branch of mathematics that deals with counting and arranging objects following certain rules. Specifically, this is a permutation problem, where we want to find the number of distinct ways to arrange 6 letters out of 26 without repetition.
To calculate this, we use the formula for permutations without repetition, which is n!/(n-r)!, where 'n' is the total number of items to choose from, 'r' is the number of items to choose, '!' denotes factorial, and 'n!/(n-r)!' denotes the number of ways to arrange 'r' items out of 'n' unique items.
Here, 'n' is 26 (letters in the alphabet) and 'r' is 6 (the number of letters on the license plate). So the calculation is 26!/(26-6)!, which simplifies to 26 x 25 x 24 x 23 x 22 x 21. This is because the factorial terms from 20! downwards on both the numerator and the denominator cancel out.
Calculating the product of those numbers gives us 26 x 25 x 24 x 23 x 22 x 21 = 165,765,600 possible license plates.
Therefore, you can make 165,765,600 unique license plates with 6 non-repeating alphabetical letters.