How many license plates can be made if each should have 3 letters of the English alphabet with no letter repeated ? what will be the answer if the letters can be repeated?
Matrix method
X-y+3z=2
-2x-2y+z=3
2x+y+2z=2
plates:
no repeated letters:
there are 26 choices for the 1st letter, leaving only 25 for the 2nd, and 24 for the 3rd: 26*25*24 = 15600
repeated letters:
26*26*26 = 17576
matrix:
If the matrix of coefficients is A, you want (x,y,z) such that
AX = B
where B = (2 3 2)
That means that X = A-1B
http://www.wolframalpha.com/input/?i=solve+{{1%2C-1%2C3}%2C{-2%2C-2%2C1}%2C{2%2C1%2C2}}*{{x}%2C{y}%2C{z}}+%3D+{{2}%2C{3}%2C{2}}
A-1:
http://www.wolframalpha.com/input/?i=inverse+{{1%2C-1%2C3}%2C{-2%2C-2%2C1}%2C{2%2C1%2C2}}
A-1B:
http://www.wolframalpha.com/input/?i=%28inverse+{{1%2C-1%2C3}%2C{-2%2C-2%2C1}%2C{2%2C1%2C2}}%29*{{2}%2C{3}%2C{2}}
To answer the first question, we need to calculate the number of ways to choose 3 letters from the English alphabet without repetition.
The English alphabet has 26 letters. For the first license plate, we have 26 choices. For the second license plate, since we cannot repeat any letters, we have 25 choices left. Similarly, for the third license plate, we have 24 choices left.
So, the total number of license plates without repeating any letters = 26 * 25 * 24 = 15,600.
As for the second question, if the letters can be repeated, then for each of the 3 positions on the license plate, we have 26 choices (since repetition is allowed). Therefore, the total number of license plates with repeated letters = 26 * 26 * 26 = 17,576.