Quiz 10
Discrete Mathematics · 2 problems · solutions hidden, click to reveal
Flashcards
Problems
Passwords have length exactly 15, composed of digits (0–9), capital letters (A–Z), and lowercase letters (a–z). (Do not simplify.)
(a) How many total passwords are there?
(b) How many contain no capital letters?
(c) How many contain at least one capital and at least one lowercase letter?
(a) How many total passwords are there?
(b) How many contain no capital letters?
(c) How many contain at least one capital and at least one lowercase letter?
Show solution
Each character has \(10 + 26 + 26 = 62\) choices.
(a) \(62^{15}\).
(b) No capitals leaves \(36\) choices each: \(36^{15}\).
(c) By inclusion–exclusion (subtract no-capital and no-lowercase, add back digits-only):
\(62^{15} - 36^{15} - 36^{15} + 10^{15}.\)
(a) \(62^{15}\).
(b) No capitals leaves \(36\) choices each: \(36^{15}\).
(c) By inclusion–exclusion (subtract no-capital and no-lowercase, add back digits-only):
\(62^{15} - 36^{15} - 36^{15} + 10^{15}.\)
There are 10 AOCs and 20 sport teams; each student declares exactly one AOC and plays exactly one team. How many students guarantee that at least three share the same AOC and same team?
Show solution
Answer: 401.
There are \(10 \cdot 20 = 200\) (AOC, team) boxes. To avoid three in any box, each holds at most \(2\), allowing \(400\) students. One more forces a box with \(3\): \(\left\lceil \tfrac{401}{200} \right\rceil = 3\). So \(401\) students suffice.