LEARN · EXPLAIN · REVISE

There are t first-stage choices with 2 continuations each, and 3 other choices with 5 continuations each. Count paths.

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

高三選擇性必修第三册(A版).pdf · 6.1 · PDF 7 / printed page 2

TOPIC 01

Addition and multiplication counting principles

Count disjoint alternatives and sequential choices without overlap or omission.

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

01 / Standard#Worked example

t=3

Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.

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

Use this intermediate relation.

N=2t+3\cdot5

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=6+15=21$
Unequal branch sizes require addition of branch products.

The requested value is 21.

Checks and common pitfalls: Unequal branch sizes require addition of branch products.

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

Teacher preparation and assessment

Count disjoint alternatives and sequential choices without overlap or omission.
Which condition is essential in addition and multiplication counting principles?
When should two counts be added rather than multiplied?

Defining relation: Count disjoint alternatives and sequential choices without overlap or omission.
$N=\sum N_i\quad\text{or}\quad N=\prod N_i$
Conditions: Addition requires disjoint cases; stage counts must hold for each branch or use separate branches.