Solution: 2019 Winter Midterm - 3
Author: Michiel SmidQuestion
Consider strings of length 15, where each character is a lowercase letter or an uppercase letter.
How many such strings contain at least one lowercase letter and at least one uppercase letter?
(a)
$52^{15} - 3 \cdot 26^{15}$
(b)
None of the above.
(c)
$52^{15} - 2 \cdot 26^{15}$
(d)
$52^{15} - 26^{15}$
Solution
A = Strings that contain at least one lowercase letter
$ \overline{A} = $ Strings that contain no lowercase letters
$ |\overline{A}| = 26^{15} $
B = Strings that contain at least one uppercase letter
$ \overline{B} = $ Strings that contain no uppercase letters
$ |\overline{B}| = 26^{15} $
$ |A \cap B| = \text{All Possibilities} - |\overline{A}| - |\overline{B}| $
$ |A \cap B| = 52^{15} - 26^{15} - 26^{15} $
$ |A \cap B| = 52^{15} - 2 \cdot 26^{15} $