Solution: 2018 Fall Midterm - 3

Author: Michiel Smid

Question

Consider strings of length 15, where each character is a lowercase letter or an uppercase letter. How many such strings contain at least two lowercase letters?

(a)

$52^{15} - 26^{15} - 15 \cdot 26^{14}$

(b)

$52^{15} - 26^{15} - 15 \cdot 26^{15}$

(c)

None of the above.

(d)

$52^{15} - 15 \cdot 26^{15}$

Solution

The size of the set of all strings of length 15 is $ 52^{15} $.

Let’s break this down into 2 cases:

A = the string contains no lowercase letters

There are $ 26^{15} $ ways to choose the characters in the string.

B = the string contains exactly one lowercase letter

There are $ 15 $ ways to choose the position of the lowercase letter.

There are $ 26 $ ways to choose the lowercase letter.

There are $ 26^{14} $ ways to choose the remaining characters.

C = The string contains at least two lowercase letters

$ |C| = |U| - |A| - |B| $

$ |C| = 52^{15} - 26^{15} - 15 \cdot 26 \cdot 26^{14} $

$ |C| = 52^{15} - 26^{15} - 15 \cdot 26^{15} $