Question

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?

Answer 1

Matrix method
X-y+3z=2
-2x-2y+z=3
2x+y+2z=2

Answer 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}}