Counting
How many positive integers are there less than 1000000 (a million), such that:
- its digits are all distinct;
- it is even;
- have exactly 3 identical digits.
Handle each condition independently.
🗝️How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other?
Interpret “way” in two different ways:
- Number of sequences of people.
- Number of arrangements of people; e.g. ‘WMMW…’, ‘MWM…’, etc.
Ninety students, including Joe and Jane, are to be split into three classes of equal size, and this is to be done at random. What is the probability that Joe and Jane end up in the same class?
Source: ItP 🗝️Twenty distinct cars park in the same parking lot every day. Ten of these cars are US-made. while the other ten are foreign-made. The parking lot has exactly twenty spaces, all in a row, so the cars park side by side. However, the drivers have varying schedules, so the position any car might take on a certain day is random.
- In how many different ways can the cars line up?
- What is the probability that on a given day, the cars will park in such a way that they alternate (no two US-made are adjacent and no two foreign-made are adjacent)?
Eight rooks are placed in distinct squares of an \(8 \times 8\) chessboard, with all possible placements being equally likely. Find the probability that all the rooks are safe from one another, i.e. that there is no row or column with more than one rook.
Source: ItP 🗝️An academic department offers 8 lower level courses: \(\lbrace L_1, L_2,\ldots, L_8\rbrace\) and 10 higher level courses: \(\lbrace H_1, H_2,\ldots, H_{10}\rbrace\). A valid curriculum consists of 4 lower level courses. and 3 higher level courses.
- How many different curricula are possible?
- Suppose that \(\lbrace H_1, \ldots, H_5\rbrace\) have \(L_1\) as a prerequisite, and \(\lbrace H_6\ldots,H_{10}\rbrace\) have \(L2\) and \(L_3\) as prerequisites. i.e.. any curricula which involve, say, one of \(\lbrace H_1, \ldots, H_5\rbrace\) must also include \(L_1\). How many different curricula are there?
How many 6-word sentences can be made using each of the 26 letters of the alphabet exactly once? A word is defined as a nonempty (possibly jibberish) sequence of letters.
Source: ItP 🗝️We draw the top 7 cards from a well-shuffled standard 52-card deck. Find the probability that:
- The 7 cards include exactly 3 aces.
- The 7 cards include exactly 2 kings.
- The probability that the 7 cards include exactly 3 aces. or exactly 2 kings, or both.
Check here the definiton of a standard deck of 52.
A standard deck has 52 cards, divided into 4 suits (hearts ♥️, diamonds ♦️, clubs ♣️, spades ♠️), each with 13 ranks (A(ce), 2, …, 10, J(ack), Q(ueen), K(ing)). The cards J, Q, and K are called face cards. Suits are broken into two colors: red and black. Each color has two suits: hearts and diamonds in red, and clubs and spades in black.