Section 01

The Difference — A Lock Code vs. A Fruit Bowl

Imagine two everyday situations. First: you’re setting a 3-digit code on a padlock using digits 1, 2, and 3. The code 1-2-3 is completely different from 3-2-1 — they open different locks (or rather, only the exact sequence you set will open yours). Order matters. This is a permutation.

Second: you’re picking 3 fruits from a bowl to make a smoothie — an apple, a banana, and an orange. It doesn’t matter if you grabbed the apple first or the orange first — you end up with the exact same smoothie ingredients either way. Order doesn’t matter. This is a combination.

Section 02

Formal Definitions — The nPr and nCr Formulas

Both formulas start from the same n items, choosing r of them — they only differ in whether order counts.

Section 03

Interactive: Permutation or Combination?

Click through 5 real scenarios. Try to guess before reading the reasoning — this is the exact judgment call you’ll need to make on every word problem you encounter.

Section 04

The Factorial Foundation

Both formulas are built entirely from factorials, so it’s worth getting comfortable with them first. n! (read “n factorial”) means multiply every whole number from n down to 1.

Section 05

Interactive: Real-Life Permutations Gallery

Four real scenarios where order matters, each fully calculated.

  Permutations in the Real World

Olympic Podium PIN Code Seating Order Password

8 sprinters race in the 100m final. How many different ways can gold, silver, and bronze be awarded?

n = 8 sprinters, r = 3 medal positions
P(8,3) = 8! / (8−3)! = 8! / 5!
= 8 × 7 × 6
= 336 possible podium outcomes

336 ways — no repetition (a sprinter can’t win two medals), order matters (gold ≠ silver ≠ bronze)

A phone lock uses a 4-digit PIN, digits 0–9, and digits CAN repeat (like 1-1-2-2). How many possible PINs exist?

10 choices for each of 4 positions
10 × 10 × 10 × 10
= 10⁴
= 10,000 possible PINs

10,000 combinations — this is a “permutation with repetition”: order matters AND digits can repeat

5 friends want a group photo, standing in a row. How many different ways can they line up?

n = 5 friends, all 5 positions filled
5! = 5 × 4 × 3 × 2 × 1
= 120 possible lineups

120 ways — this is the special case where r = n (arranging ALL the items), so it’s just n!

A password requires exactly 4 DIFFERENT letters (no repeats), chosen from the 26-letter alphabet. How many such passwords exist?

n = 26 letters, r = 4 positions, no repeats
P(26,4) = 26! / (26−4)! = 26! / 22!
= 26 × 25 × 24 × 23
= 358,800 possible passwords

358,800 ways — order matters (ABCD ≠ DCBA) and no letter can repeat

Section 06

Interactive: Real-Life Combinations Gallery

Four real scenarios where order doesn’t matter — only which items get chosen.

  Combinations in the Real World

Lottery Committee Pizza Toppings Poker Hand

A lottery draws 6 numbers from a pool of 49. How many different winning combinations are possible?

n = 49 numbers, r = 6 drawn
C(49,6) = 49! / [6! × 43!]
= (49×48×47×46×45×44) / 720
= 13,983,816 possible combinations

~14 million combinations — {1,2,3,4,5,6} is the same winning ticket regardless of the order the balls were drawn in

A club with 10 members needs to select a 3-person committee, with no distinct roles. How many possible committees exist?

n = 10 members, r = 3 chosen
C(10,3) = 10! / [3! × 7!]
= (10×9×8) / (3×2×1)
= 720 / 6 = 120 possible committees

120 ways — {Alice, Bob, Carla} is one single committee, regardless of who was “picked first”

A pizza place offers 8 toppings. How many different 3-topping pizzas can you order?

n = 8 toppings, r = 3 chosen
C(8,3) = 8! / [3! × 5!]
= (8×7×6) / (3×2×1)
= 336 / 6 = 56 possible pizzas

56 unique pizzas — a mushroom-olive-pepper pizza is identical to a pepper-mushroom-olive pizza

A standard deck has 52 cards. How many different 5-card poker hands are possible?

n = 52 cards, r = 5 dealt
C(52,5) = 52! / [5! × 47!]
= (52×51×50×49×48) / 120
= 2,598,960 possible hands

~2.6 million hands — the order cards were dealt in doesn’t change what hand you’re holding

Section 07

The “Does Order Matter?” Test

A reliable 3-step process for any word problem you encounter:

Section 08

Worked Example — Building a Real Password System

Let’s combine everything into one realistic problem: a website requires passwords with exactly 2 different letters (no repeats) followed by 3 digits (digits CAN repeat). How many total passwords are possible?

Section 09

Permutations and Combinations with Repetition

The formulas above assume each item can only be chosen once. Real life often allows repeats — here’s how the math changes.

Worked Example — Ice Cream Scoops

An ice cream shop has 5 flavors. You order 3 scoops, and you’re allowed to repeat a flavor (like 2 scoops chocolate + 1 scoop vanilla). Order doesn’t matter — a cup with chocolate-chocolate-vanilla is the same order regardless of which scoop went in first.

Section 10

Pascal’s Triangle Connection

Every entry in Pascal’s Triangle IS a combination value. Row n, position k gives you exactly C(n,k) — no calculation required, just count and look it up.

Look at row n=6: the entries are C(6,0)=1, C(6,1)=6, C(6,2)=15, C(6,3)=20, and so on. Each number is the sum of the two numbers diagonally above it — which is exactly Pascal’s Rule: C(n,k) = C(n−1,k−1) + C(n−1,k).

Section 11

Real-World Domains

Cryptography

Key space size for a cipher is a permutation calculation — the number of possible keys directly determines how brute-forceable an encryption scheme is.

Genetics

Counting possible DNA sequences, or the number of ways alleles can combine in offspring, relies directly on permutation and combination formulas.

Tournament Brackets

The number of possible ways a single-elimination bracket can play out, or how many unique round-robin schedules exist, is pure combinatorics.

Scheduling

Assigning employees to shifts, or students to exam time slots, is a permutation/combination problem — especially when constraints (no repeats, fixed roles) apply.

Quality Control

Choosing a random sample of r items from a batch of n to inspect for defects is a classic combination — order of inspection doesn’t matter.

Game Design

Loot drop tables, card game deck compositions, and probability balancing in games all rely on combinatorics to calculate fair odds.

Section 12

Code Examples — Python

Using math.perm() and math.comb() (Python 3.8+)

import math

# Olympic podium — 3 medals from 8 finalists, order matters
print(math.perm(8, 3))     # 336

# Lottery — choose 6 numbers from 49, order doesn't matter
print(math.comb(49, 6))    # 13983816

# Password letters — 2 of 26, no repeats, order matters
print(math.perm(26, 2))    # 650

# Pizza toppings — choose 3 of 8, order doesn't matter
print(math.comb(8, 3))     # 56

# PIN code — permutation WITH repetition (not built into math module)
pin_combos = 10 ** 4
print(pin_combos)          # 10000

# Ice cream scoops — combination WITH repetition (stars and bars)
ice_cream = math.comb(5 + 3 - 1, 3)
print(ice_cream)          # 35

Generating the actual arrangements with itertools

from itertools import permutations, combinations

runners = ['A', 'B', 'C', 'D']

# All ordered podium outcomes (top 3 of 4 runners)
podiums = list(permutations(runners, 3))
print(len(podiums))      # 24  =  4P3
print(podiums[:3])
# [('A','B','C'), ('A','B','D'), ('A','C','B'), ...]

# All unordered committees (any 3 of 4 people)
committees = list(combinations(runners, 3))
print(len(committees))   # 4  =  4C3
print(committees)
# [('A','B','C'), ('A','B','D'), ('A','C','D'), ('B','C','D')]

Section 13

Knowledge Quiz

Click a question to expand it, then pick your answer.

Q4 — Why is choosing a 5-card poker hand a combination rather than a permutation?

Because there are only 52 cards in a deck Because face cards are different from number cards Because the order the cards were dealt in doesn’t change which hand you’re holding Because repetition is allowed in card games

A hand of {A, K, 7, 3, 2} is the same hand no matter which card was dealt first. Since order doesn’t affect the outcome, it’s counted with combinations: C(52,5) = 2,598,960.