LEARN · EXPLAIN · REVISE

Arrange m distinct people in a line with A and B adjacent.

Read the idea, work independently, then explain what changed.

高三選擇性必修第三册(A版).pdf · 6.2 · PDF 19 / printed page 14

TOPIC 01

Permutations and combinations

Distinguish ordered selections, unordered sets and symmetry in arrangements.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

m=9

Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Decide whether choices are alternative cases or successive stages, then check overlap.

Hint 2

Treat A,B as one block.

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=2(m-1)!=80640$
Multiply by two for AB and BA inside the block.

The requested value is 80640.

Checks and common pitfalls: Multiply by two for AB and BA inside the block.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Defining relation: Distinguish ordered selections, unordered sets and symmetry in arrangements.
$P_n^r=n!/(n-r)!;\quad C_n^r=n!/[r!(n-r)!]$
Conditions: Counts use nonnegative integer sizes; decide whether labels, order and rotations distinguish outcomes.

Dividing by r! is valid only when every selected set has exactly r! counted orders.