“Order matters, P; order does not, C”
Choosing between the two
A password is a permutation because order changes it. A committee is a combination because it does not. Ask that one question first, every time.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Permutations, Combinations and Binomial Theorem formulas
These two chapters share one skill: deciding whether order matters. Once that is settled the formulas are short, and the binomial theorem then rests on the same combination symbol.
17 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Counting
| What | Formula | Symbols and notes |
|---|---|---|
| Fundamental principle | if one task has m ways and the next has n, together they have mn | multiply along, add between alternatives |
| Permutations | nPr = n!/(n − r)! | order matters, as in arranging |
| Combinations | nCr = n!/(r!(n − r)!) | order does not matter, as in selecting |
| Relation | nPr = nCr × r! | arrange after selecting |
| Symmetry | nCr = nC(n−r) | choosing r to keep is choosing n − r to leave |
| Pascal's rule | nCr + nC(r−1) = (n+1)Cr | the rule that builds Pascal's triangle |
Standard arrangements
| What | Formula | Symbols and notes |
|---|---|---|
| Circular permutations | (n − 1)! | one position is fixed to remove rotations |
| Necklace or garland | (n − 1)!/2 | a flip gives the same arrangement |
| With repeated items | n!/(p! q! r!) | p, q, r are the counts of each repeated item |
| Repetition allowed | nr | r places, n choices each time |
| At least one | total − none | almost always faster than counting cases |
Binomial theorem
| What | Formula | Symbols and notes |
|---|---|---|
| Expansion | (a + b)n = Σ nCr an−r br | n + 1 terms in all |
| General term | Tr+1 = nCr an−r br | note the r + 1, which trips most students |
| Middle term, n even | the (n/2 + 1)th term | one middle term |
| Middle term, n odd | the (n+1)/2 th and (n+3)/2 th terms | two middle terms |
| Sum of coefficients | put a = b = 1, giving 2n | a one-line trick |
| Greatest coefficient | the middle term's | for a symmetric expansion |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Circle loses one”
Circular arrangements
Around a table there is no first seat, so fix one person and arrange the rest: (n − 1)!. A garland halves it again because it can be flipped over.
“r plus one”
The general term of a binomial expansion
Tr+1 uses r, so the fourth term needs r = 3. Half the mistakes in this chapter are an off-by-one here.
“At least one means all minus none”
Counting problems
Counting the unwanted case and subtracting is nearly always shorter than counting the wanted ones.
Where marks go
The mistakes that cost the most.
- Using permutations for a selection where order does not matter, which inflates the answer by r!.
- Forgetting to divide by the factorials of repeated letters when arranging a word.
- Using n! for a circular arrangement instead of (n − 1)!.
- Taking r as the term number in the binomial general term. The (r + 1)th term uses r.
- Missing that an odd power gives two middle terms, not one.
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