Lecture 15 — Counting, Combinatorics, and Choice

Learning objectives

• Apply the multiplication and addition principles
• Compute combinations and permutations from first principles
• Justify over-counting corrections

Case-based lecture: you attempt, we discuss, then we reveal.

The plan

• Four cases: cs-057, cs-058, cs-133, cs-059, cs-060
• Each: 5-minute attempt → discussion → reveal → prompt
• Worked examples in between — ask anything
• Break mid-lecture · exit question at the end

Key definitions

• Factorial — n! = n x (n-1) x ... x 1; the number of ways to arrange n items in order
• Permutation — an ordered selection; order matters, so AB and BA are different
• Combination — an unordered selection; order does not matter, so AB and BA are the same
• Complement counting — count the opposite of what you want, then subtract from the total
• Pigeonhole principle — if more than n items go into n boxes, some box holds two
• Binomial coefficient — C(n, k) = n! / (k! (n-k)!); the number of k-sized teams from n people

The locker problem: divisors leave squares open

open_lockers = [False] * 101
for student in range(1, 101):
    for locker in range(student, 101, student):
        open_lockers[locker] = not open_lockers[locker]
squares = [n for n in range(1, 101) if open_lockers[n]]
print(squares[:5], squares[-1], len(squares))

Expected: open lockers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Cost: Simulation O(n log n) (harmonic sum); the mathematical answer is O(1) - reasoning beats brute force.

Team photo: count first, then block

import math
total = math.factorial(5)
blocked = math.factorial(4) * 2
print(total, blocked, total - blocked)

Expected: 72

Cost: O(1) with the counting argument; enumerating 120 lineups is fine too but does not scale.

The committee draw: complements and stars

import math
c = math.comb
total = c(10, 4)
forbidden = c(6, 4) + c(4, 4)
print(total, forbidden, total - forbidden)

Expected: 194

Cost: O(1) arithmetic via factorials; a brute-force enumeration would touch all 210 committees.

The birthday question: product of complements

p = 1.0
n = 0
while p >= 0.5:
    n += 1
    p *= (366 - n) / 365
print(n, round(1 - p, 3))

Expected: 23 people (P(shared) ~ 50.7%)

Cost: O(answer) for the product, O(trials x n) for the simulation - both trivial here.

• **Input:** 100 lockers, 100 students; student i toggles lockers divisible by i
• **Output:** open lockers are perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-057 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"When is enumeration a legitimate answer rather than a crutch?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-057 — not embedded in this deck.

• **Input:** 5 students; students P and Q refuse to stand together
• **Output:** 5! = 120; together (block): 4! x 2 = 48; apart: 120 - 48 = 72
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-058 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Where exactly did the wrong counts double-count — point at the pair."**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-058 — not embedded in this deck.

• **Input:** the schedule, loop legs, fleet size
• **Output:** `departures 0, 10, 25, 35, 50, 60` — gaps `10, 15, 10, 15,
• 10` — verdict + notice
• Solve for the worked instance first. Then ask what would change if one

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-133 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Operations proposes shortening the loop by three minutes instead of buying a third bus. Does the poster claim become true? Show the departure ledger that decides it."**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-133 — not embedded in this deck.

• **Input:** 6 CS, 4 DS; committee of 4 with at least one of each programme
• **Output:** total C(10,4)=210; all-CS C(6,4)=15; all-DS C(4,4)=1; valid 210-16 = 194
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-059 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"When is enumeration a legitimate answer rather than a crutch?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-059 — not embedded in this deck.

• **Input:** n people, 365 equally likely birthdays (ignore leap days)
• **Output:** P(no shared) = 365/365 x 364/365 x ... ; n=23 gives P(shared) ~ 50.7%
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-060 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Where exactly did the wrong counts double-count — point at the pair."**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-060 — not embedded in this deck.

Common misconceptions

• Multiplying when you should add: 'and' stages multiply, 'or' cases add.
• C(10,4) computed as 10x9x8x7 - the order of a chosen team was counted 4! times.
• Applying complement counting with overlapping cases - subtraction needs disjoint sets.
• Believing the pigeonhole principle needs large numbers - 3 socks, 2 colours is already enough.

Summary & exit question

• Named technique today: Counting, Combinatorics, and Choice
• Four cases attempted — methods before answers

Exit question (choose one, 60 seconds)

1. When is the complement trick the right tool?
2. Permutations vs combinations in one phrase each.
3. How does simulation back up counting?