Permutation and Combinations: Concepts, Formulas, and Clear Examples

Permutation ($nPr$) counts ordered arrangements; combination ($nCr$) counts unordered selections. Decision rule: if swapping people/seats changes the outcome, it is permutation.

Campus MCQs love circular permutation, restrictions (together/apart), and at least/at most via complement counting.

Keep $nCr = nC(n-r)$ and small-Pascal rows handy for sanity checks.

Factorial and notation

Permutation

$$^nP_r = \frac{n!}{(n-r)!}$$

Combination

$$^nC_r = \frac{n!}{r!(n-r)!}$$

Relation

P vs C

$$^nP_r = r! \cdot {}^nC_r$$

Example — Choose committee

Choose 3 from 10: ${}^{10}C_3 = \frac{10\cdot9\cdot8}{3\cdot2\cdot1}=120$.

Circular permutation

Circle

$$(n-1)!$$

Fix one person to break rotation.

Necklace (flip)

$$\frac{(n-1)!}{2}$$

Restrictions

• All together: treat block as one unit, multiply internal $r!$.
• No two together: gap method after placing others.

Complement counting

• At least one = total − none.
• At most k: sum combinations for 0..k.

Identical objects

Multiset permutations

$$\frac{n!}{n_1! n_2! \cdots}$$

Step-by-step method for exam questions

1. Decide order matters or not.
2. Handle restrictions via blocks or complement.
3. Write $nPr$/$nCr$ formula; cancel factorials early.
4. For circular, apply $(n-1)!$ rule.
5. Numeric simplify before multiplying huge factorials.
6. Match option.

Formula cheat sheet

Frequently asked questions

0! value?

1 by definition.

nC0?

1 (empty selection).

Digits with leading zero?

Often forbidden—reduce first digit choices.

Same objects?

Divide by factorial of identical groups.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.