LEARN · EXPLAIN · REVISE

Counting principles: mixed review

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

TOPIC 01

Counting principles: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

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

01 / Foundation#Your turn

Choose one book: t mathematics titles or 3 history titles, with no overlap. How many choices?

t=20

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=t+3

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=20+3=23$
The two subject cases are disjoint alternatives.

The requested value is 23.

Checks and common pitfalls: The two subject cases are disjoint alternatives.

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

02 / Foundation#Your turn

Choose an unordered committee of 3 from m people.

m=15

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.

C_m^3=P_m^3/3!

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=15(14)(13)/6=455$
Each committee was counted in six possible orders.

The requested value is 455.

Checks and common pitfalls: Each committee was counted in six possible orders.

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

03 / Foundation#Your turn

Find the sum of all coefficients of (2+x)^m.

m=4

The exponent n is a nonnegative integer; term number is r+1, not r.

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

Substitute x=1.

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$(2+1)^{4}=81$
Use x=1 for the full coefficient sum, not 2^m unless both base coefficients equal one.

The requested value is 81.

Checks and common pitfalls: Use x=1 for the full coefficient sum, not 2^m unless both base coefficients equal one.

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

04 / Foundation#Your turn

Among t+8 students, t study art, 8 study music and 3 study both. How many study either subject?

t=23

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.

|A\cup B|=|A|+|B|-|A\cap B|

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=23+8-3=28$
Subtract the overlap once because it was counted twice.

The requested value is 28.

Checks and common pitfalls: Subtract the overlap once because it was counted twice.

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

05 / Foundation#Your turn

Choose a president and secretary from m distinct people. Count assignments.

m=7

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.

P_m^2=m(m-1)

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=7(7-1)=42$
The two offices are different, so order matters.

The requested value is 42.

Checks and common pitfalls: The two offices are different, so order matters.

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

06 / Foundation#Your turn

Find the coefficient of x² in (1+x)^m.

m=7

The exponent n is a nonnegative integer; term number is r+1, not r.

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.

C_m^2=m(m-1)/2

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$[x^2]=21$
The unordered pair of chosen factors determines the term.

The requested value is 21.

Checks and common pitfalls: The unordered pair of chosen factors determines the term.

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

07 / Standard#Your turn

Among t+8 students, t study art, 8 study music and 3 study both. How many study either subject?

t=26

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.

|A\cup B|=|A|+|B|-|A\cap B|

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=26+8-3=31$
Subtract the overlap once because it was counted twice.

The requested value is 31.

Checks and common pitfalls: Subtract the overlap once because it was counted twice.

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

08 / Standard#Your turn

From m people, form a 3-person committee that includes a specified person.

m=10

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.

C_{m-1}^2

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=C_{9}^2=36$
Select only the remaining two members from the other people.

The requested value is 36.

Checks and common pitfalls: Select only the remaining two members from the other people.

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

09 / Standard#Your turn

Find the alternating sum of binomial coefficients for m≥1.

\sum_{r=0}^{10}(-1)^rC_{10}^r

The exponent n is a nonnegative integer; term number is r+1, not r.

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

Substitute x=-1 in (1+x)^m.

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$(1-1)^{10}=0$
The exponent is positive, so the substitution is unambiguous.

The requested value is 0.

Checks and common pitfalls: The exponent is positive, so the substitution is unambiguous.

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

10 / Standard#Your turn

Among t+8 students, t study art, 8 study music and 3 study both. How many study either subject?

t=29

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.

|A\cup B|=|A|+|B|-|A\cap B|

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=29+8-3=34$
Subtract the overlap once because it was counted twice.

The requested value is 34.

Checks and common pitfalls: Subtract the overlap once because it was counted twice.

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

11 / Standard#Your turn

From m people, form a 3-person committee that includes a specified person.

m=13

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.

C_{m-1}^2

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=C_{12}^2=66$
Select only the remaining two members from the other people.

The requested value is 66.

Checks and common pitfalls: Select only the remaining two members from the other people.

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

12 / Standard#Your turn

Find the alternating sum of binomial coefficients for m≥1.

\sum_{r=0}^{13}(-1)^rC_{13}^r

The exponent n is a nonnegative integer; term number is r+1, not r.

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

Substitute x=-1 in (1+x)^m.

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$(1-1)^{13}=0$
The exponent is positive, so the substitution is unambiguous.

The requested value is 0.

Checks and common pitfalls: The exponent is positive, so the substitution is unambiguous.

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

13 / Transfer#Your turn

Passwords of length 3 use t symbols with repetition allowed. Exclude those with all symbols identical. Count remaining passwords.

t=32

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=t^3-t

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$N=32768-32=32736$
The excluded all-identical passwords are one per symbol, not one total.

The requested value is 32736.

Checks and common pitfalls: The excluded all-identical passwords are one per symbol, not one total.

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

14 / Transfer#Your turn

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

m=5

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.
$N=2(m-1)!=48$
Multiply by two for AB and BA inside the block.

The requested value is 48.

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

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

15 / Transfer#Your turn

Find the alternating sum of binomial coefficients for m≥1.

\sum_{r=0}^{5}(-1)^rC_{5}^r

The exponent n is a nonnegative integer; term number is r+1, not r.

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

Substitute x=-1 in (1+x)^m.

Worked solution

Decide whether choices are alternative cases or successive stages, then check overlap.
Apply the stated relation and retain its conditions.
$(1-1)^{5}=0$
The exponent is positive, so the substitution is unambiguous.

The requested value is 0.

Checks and common pitfalls: The exponent is positive, so the substitution is unambiguous.

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

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.