Dr. Samir Orujov
School of Business (ADA)
September 2025
Master the fundamental counting principle for multi-step problems
Calculate arrangements where order matters using permutation formulas
Determine selections where order doesn't matter using combination formulas
Understand and apply binomial expansions with combinatorial proofs
Calculate complex arrangements with multiple categories
Solve problems using stars and bars methodology
If experiment 1 has $m$ outcomes and experiment 2 has $n$ outcomes:
This extends to any number of sequential experiments.
Letters: $26^3 = 17,576$ ways
Numbers: $10^4 = 10,000$ ways
Total: $17,576 \times 10,000 = 175,760,000$
Problem: Restaurant has 4 appetizers, 6 main courses, 3 desserts. How many complete meals?
Answer: $4 \times 6 \times 3 = 72$ complete meals
Permutation: An arrangement of objects where order is important
Number of ways to arrange $r$ objects from $n$ distinct objects
$n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1$
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
$0! = 1$ (by definition)
Letters: P-E-P-P-E-R (6 total)
Repetitions: P appears 3 times, E appears 2 times, R appears 1 time
Problem: How many ways to arrange 9 players in batting order?
Answer: $9! = 362,880$ arrangements
Combination: A selection of objects where order is not important
Number of ways to select \(r\) objects from \(n\) distinct objects
Permutation: ABC, ACB, BAC are different
Combination: ABC, ACB, BAC are the same
Therefore: \(C(n,r) = \dfrac{P(n,r)}{r!}\)
Problem: Select 3 people from 20 for a committee
Problem: From 20 people, select 5 for a committee, but 2 people refuse to serve together.
Answer: Total − Both included = \(C(20,5)-C(18,3)=15,504-816=14,688\)
Expands any binomial expression raised to a power
$(x+y)^3 = \binom{3}{0}x^0y^3 + \binom{3}{1}x^1y^2 + \binom{3}{2}x^2y^1 + \binom{3}{3}x^3y^0$
$= 1 \cdot 1 \cdot y^3 + 3 \cdot x \cdot y^2 + 3 \cdot x^2 \cdot y + 1 \cdot x^3 \cdot 1$
$= y^3 + 3xy^2 + 3x^2y + x^3$
The coefficients $\binom{n}{k}$ form Pascal's Triangle
Key Insight: Each term comes from choosing $x$ or $y$ from $n$ factors
$(x+y)(x+y)(x+y) = $ ?
Click "Build Terms" to see all possible combinations!
Use the Pascal's Triangle generator to explore binomial coefficients!
Multinomial Coefficient: Arrangements when objects fall into multiple categories
Where $n_1 + n_2 + ... + n_r = n$
Problem: Assign 10 officers to 3 shifts
Day shift: 5 officers, Evening: 2 officers, Night: 3 officers
Problem: 12 teams, divide into 3 groups of 4
Problem: Divide 15 students into groups of 6, 5, and 4. How many ways?
Answer: $\binom{15}{6,5,4} = \frac{15!}{6! \cdot 5! \cdot 4!} = 630,630$
Where the sum is over all non-negative integers $n_1, n_2, ..., n_r$ such that $n_1 + n_2 + ... + n_r = n$
Step 1: Find all ways to write $2 = n_1 + n_2 + n_3$
$(2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), (0,1,1)$
Step 2: Calculate coefficients and terms
Build terms for $(x_1 + x_2 + x_3)^2$:
From each factor, choose one variable:
$(x_1 + x_2 + x_3)^2 = (x_1 + x_2 + x_3) \times (x_1 + x_2 + x_3)$
Factor 1: $(x_1 + x_2 + x_3)$
Factor 2: $(x_1 + x_2 + x_3)$
Choose one variable from each factor:
Current term: Select variables above
Click "Show All Terms" to see the complete expansion:
Use the tools above to construct your understanding of multinomial expansion!
Problem Type: Find non-negative integer solutions to $x_1 + x_2 + ... + x_r = n$
Think of it as: Distributing $n$ identical objects into $r$ distinguishable boxes
Method:
Example: $x_1 + x_2 + x_3 = 5$ (find non-negative solutions)
Stars: ⭐⭐⭐⭐⭐ (represent the sum)
Bars: | | (separate into 3 groups)
Arrangement: ⭐⭐|⭐|⭐⭐ means $x_1=2, x_2=1, x_3=2$
⭐⭐⭐⭐⭐|| → $x_1=5, x_2=0, x_3=0$
||⭐⭐⭐⭐⭐ → $x_1=0, x_2=0, x_3=5$
⭐|⭐⭐|⭐⭐ → $x_1=1, x_2=2, x_3=2$
Problem: Invest \$20,000 among 4 different investments (in \$1,000 units)
Find number of ways to distribute: $x_1 + x_2 + x_3 + x_4 = 20$
Minimum investments: Each investment ≥ $2,000
Solution: Substitute $y_i = x_i - 2$, solve $y_1 + y_2 + y_3 + y_4 = 12$
Answer: $\binom{15}{3} = 455$ strategies
For positive solutions: Use substitution $y_i = x_i + 1$
For minimum requirements: Subtract minimums, then apply formula
Key insight: Transform constrained problems into standard form
Problem: Distribute 15 units among 3 investments, each getting at least 1 unit.
Answer: $\binom{(15-3)+3-1}{3-1} = \binom{14}{2} = 91$ ways
A code has 3 letters followed by 2 digits. How many possible codes?
How many ways can 5 people sit in a row of 5 chairs?
From 12 people, select a committee of 4. How many ways?
What is the coefficient of $x^2y^3$ in $(x+y)^5$?
Divide 9 objects into groups of 4, 3, and 2. How many ways?
$n_1 \times n_2 \times \cdots \times n_r$
Sequential choices$P(n,r) = \frac{n!}{(n-r)!}$
Order matters$C(n,r) = \frac{n!}{r!(n-r)!}$
Order doesn't matter$(x+y)^n$
$=\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$
Expansions$\frac{n!}{n_1! \cdot n_2! \cdots n_k!}$
Multiple groups$\binom{n+r-1}{r-1}$
Stars and barsIn how many ways can 6 people sit around a circular table?
Correct! For circular arrangements, we fix one person and arrange the rest: (n-1)!
Click "Start Quiz" to begin
Create a real-world problem using 3+ techniques from today
Multiplication principle for multi-step processes
Arrangements with and without repetition
Selections where order doesn't matter
Expansions and Pascal's triangle connections
Multiple categories and expansions
Stars and bars for distribution problems
Combinatorial Analysis Mastery Complete
Advanced Mathematical Thinking
Work through textbook problems
Link to probability & statistics
Use in real project scenarios
Advanced topics & applications
What problems would you like to solve with these new skills?
Feel free to reach out for further discussion!