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.

Week map

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!.

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

Deep study

Statistics I · Week 6 — Permutations and combinations

Week map

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 $n$.

Combination notation

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

Decision rule

Pattern families

Easy — Small n compute directly

Medium — Formula application

Hard — Complement or restrictions

Worked mini-examples

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)

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

Pattern families

Easy — Order test

Medium — Compute nCr

Hard — Multi-step select

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

1. Difference permutation vs combination?
2. Formula for nCr?
3. When is order important?

Practice loop

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

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk