LEARN · EXPLAIN · REVISE

Place t identical balls in 3 labeled boxes; empty boxes are allowed. Count distributions.

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 / Standard#Your turn

t=11

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

Use this intermediate relation.

x_1+x_2+x_3=t,\quad x_i\ge0

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=C_{13}^2=78$
Stars and bars count integer allocations; labeled boxes remain distinct.

The requested value is 78.

Checks and common pitfalls: Stars and bars count integer allocations; labeled boxes remain distinct.

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.