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Statistics I · Week 6 — Permutations & combinations

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Statistics I · Week 6 — Permutations & combinations

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2026-08-16

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Now · Week map

Week 6 — permutations & combinations

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Order matters? → permutation → combination → formula pick

Classify → Represent → Execute → Trap-check

Recognize: Ask: Difference permutation vs combination?
Procedure: Ask: would ABC differ from BAC? Yes → permutation. No → combination.
Variations / traps: Watch for: Permutation when order irrelevant.

Formula chain (compressed)

order matters? → nPr = n!/(n−r)! → nCr = nPr/r!.

Permutation — nPr = n!/(n−r)! — order matters
Combination — nCr = n!/(r!(n−r)!) — order irrelevant
Identity — nCr = nC(n−r) — symmetry trick
Multiset — divide by duplicate factorials — repeated letters
Choose vs arrange — combination then arrange? — read problem wording

Open interactive formula desk · Week 6 tab.

Deep study

Statistics I · Week 6 — Permutations and combinations

Deep study for Quiz 2 week 6. Permutations count ordered arrangements; combinations count unordered selections.

Week map

Order matters? → permutation → order irrelevant? → combination → formulas → complement counting → word-problem translation.

Permutation notation

$P(n, r)$ or ${}^nP_r$ → permutations of $r$ items chosen from $n$ distinct → order matters.
Formula: $P(n,r) = \dfrac{n!}{(n-r)!}$ .
Special case: $P(n,n) = n!$ — arrange all $n$ .

Mini-example: Top 2 finishers from 5 runners. Order matters (gold vs silver):

P(5,2) = 5 \times 4 = 20

Combination notation

$C(n, r)$ or $\binom{n}{r}$ → combinations — choose $r$ from $n$ → order does not matter.
Formula: $C(n,r) = \dfrac{n!}{r!(n-r)!}$ .
$\binom{n}{0} = 1$ ; $\binom{n}{n} = 1$ .

Mini-example: Choose 2 toppings from 5 regardless of order:

C(5,2) = \frac{5 \times 4}{2 \times 1} = 10

Decision rule

Question phrasing	Use
Arrange, rank, sequence, PIN order	Permutation
Choose, select, committee, hand	Combination

Pattern families

Easy — Small n compute directly

List or multiply descending factors:

5 \times 4 \times 3

for

P(5,3)

. Divide by

r!

when order ignored.

Medium — Formula application

Plug into

P(n,r)

C(n,r)

. Simplify factorials before multiplying large numbers.

Hard — Complement or restrictions

“At least one” → total minus “none.” Adjacent or fixed-position constraints may reduce to smaller permutation subproblem.

Worked mini-examples

Example 1 — Permutation.

P(4,2) = 4 \times 3 = 12

Example 2 — Combination.

\binom{6}{2} = \frac{6 \times 5}{2} = 15

Example 3 — Compare.

3-letter code from {A,B,C,D} without repeat:

P(4,3) = 24

. Choose 3 letters unordered:

\binom{4}{3} = 4

Example 4 — Committee.

Choose 3 from 7 people for roles-free committee:

\binom{7}{3} = 35

Example 5 — Complement.

Choose at least one red from 3 red and 4 blue (choose 2 total): total

\binom{7}{2}=21

minus no red

\binom{4}{2}=6

→ 15.

Traps

Using combination when order matters (rankings, passwords).
Using permutation when order irrelevant (teams, subsets).
Forgetting to divide by $r!$ when converting ordered to unordered.
$C(n,r) = C(n, n-r)$ symmetry — useful check.
Identical items need division beyond basic $C(n,r)$ (not all distinct).

Diagnostic (try yourself)

How many ways to arrange 3 distinct books on a shelf?
How many 2-person committees from 6 people (order irrelevant)?
Compute $P(5,3)$ and $C(5,3)$ . Which is larger?
Word problem: “President and VP from 8 candidates.” Permutation or combination? Count?
$\binom{10}{0}$ equals what?

ChatGPT prep archive

Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

Permutation: order matters; nPr = n!/(n−r)!.
Combination: order irrelevant; nCr = n!/(r!(n−r)!).
Choose r from n: if roles differ, permutation; if committee, combination.
With repetition variants change formulas—read problem.

Notation & vocabulary

Symbol	Meaning
nPr	permutations of r from n
nCr	combinations of r from n
$\binom{n}{r}$	same as nCr

Pattern families

Easy — Order test

Ask: would ABC differ from BAC? Yes → permutation. No → combination.

Medium — Compute nCr

Cancel factorial factors before multiplying large numbers. Symmetry nCr = nC(n−r).

Hard — Multi-step select

Choose committee with roles: multiply combinations/permutations per slot. Ensure no double count when constraints overlap.

Drill these on the pattern atlas — filter to week 6.

Traps

Permutation when order irrelevant.
Double counting identical committees.
r > n giving nonzero wrongly.
Ignoring indistinguishable items.

Retrieval prompts

Difference permutation vs combination?
Formula for nCr?
When is order important?

Practice loop

Read Deep study (if present) or core concepts once.
Recite the formula chain without looking.
Open one easy pattern on the interactive atlas for week 6.
Attempt without solutions; mark studied after an honest try.
Say one trap aloud before closing the tab.