📝 Chapter 2: Permutations

Evaluate \( 7! - 5! \).

Compute \( P(9,3) \).

How many distinct arrangements of the letters of "BANANA"?

In how many ways can 9 distinct people sit around a circular table?

Solve for \( n \): \( P(n,2) = 56 \).

A teacher arranges 4 girls and 3 boys in a row. How many arrangements if all girls must sit together?

A 5-character password uses any letter (26) followed by any digit (10), repeated, with no other restriction. How many passwords have exactly 3 letters then 2 digits, with letters distinct and digits distinct?

In how many ways can the letters of "ALGEBRA" be arranged so that the word starts and ends with a vowel? Show your case analysis.

Determine the number of ways 5 women and 4 men can sit in a row if the men cannot sit next to each other. (Hint: place the women first, then place men in the gaps.)

In how many ways can 4 couples sit around a circular table if each couple must sit together? Show your reasoning, including how circular permutations interact with the "glue" trick.

Solve algebraically: \( P(n+1, 2) = 30 \). Show all work and reject any extraneous solutions.

A student claims that the number of arrangements of the letters of "TOOL" is \( 4! = 24 \). Is this correct? Explain the error and provide the correct value.

Explain in your own words why the number of circular arrangements of \( n \) distinct objects is \( (n-1)! \) and not \( n! \). Use a small example (e.g., \( n = 4 \)) to support your explanation.

Describe a step-by-step procedure for solving a "permutations with restrictions" problem (e.g., "must be adjacent" or "cannot be adjacent"). Include both the "glue method" and the "complement / gaps" method.

Compare a permutation and a combination. Give an example of each, and explain how the formulas \( P(n,r) = \dfrac{n!}{(n-r)!} \) and \( C(n,r) = \dfrac{n!}{r!(n-r)!} \) are related.

Ontario licence plates have the form ABC 123 (3 letters, then 3 digits). a) How many plates are possible if all are allowed to repeat? b) How many start with the letter "M"?

A robot must walk on a grid from \( (0,0) \) to \( (5,3) \), making only right (R) and up (U) moves. How many distinct paths are possible? (This is a permutation-with-identical-objects problem.)

A baseball coach must select a batting order for 9 players. If two specific players, the captain and the lead-off hitter, must bat 1st and 2nd respectively, how many batting orders are possible?

A 5-digit password uses digits 0–9. How many passwords are there with: a) all distinct digits, b) at least one repeated digit?