JEE Maths›Permutations & Combinations›Common Mistakes

Common Mistakes

Traps in Permutations & Combinations

5 mistake patterns students fall for. 3 high-frequency traps appear in almost every exam.

Confusing permutation and combination

Very CommonFORMULA

Using ${}^nP_r$ when order does not matter or ${}^nC_r$ when it does.

Why: The question does not always explicitly say 'arrangement' or 'selection'.

WRONG: Selecting a committee of 3 from 10:

{}^{10}P_3 = 720

RIGHT: Selection (no order):

{}^{10}C_3 = 120

. Use

{}^nP_r

only when arrangement matters.

See pattern: Number Formation →

Overcounting in distribution

Very CommonFORMULA

Not dividing by $k!$ when distributing into identical groups of equal size.

Why: Identical groups are interchangeable, so swapping groups does not create a new arrangement.

WRONG: Split 6 people into 2 groups of 3:

{}^6C_3 = 20

RIGHT:

{}^6C_3 / 2! = 10

(groups are identical, so

\{A,B,C\}/\{D,E,F\} = \{D,E,F\}/\{A,B,C\}

)

See pattern: Distribution Problems →

Circular vs linear arrangement

Very CommonFORMULA

Using $n!$ instead of $(n-1)!$ for circular arrangements.

Why: In a circle, rotations of the same arrangement are identical, so we fix one element.

WRONG: 8 people around a circular table:

8!

ways

RIGHT: Fix one person, arrange the rest:

(8-1)! = 7! = 5040

ways.

See pattern: Word Arrangements →

Forgetting the leading zero constraint

CommonCASE MISS

When forming $n$ -digit numbers, the first digit cannot be 0.

Why: A number like 0123 is not a 4-digit number; it is 123 (3 digits).

WRONG: 4-digit numbers from

\{0,1,2,3\}

4! = 24

RIGHT: First digit: 3 choices (not 0). Remaining:

3! = 6

. Total:

3 \times 6 = 18

Missing cases in constrained counting

CommonCASE MISS

Not considering all valid partitions or configurations when constraints are involved.

Why: Complex constraints often split into multiple sub-cases that are easy to overlook.

WRONG: Only considering one way to split when multiple distributions are possible

RIGHT: List all valid partitions/cases systematically, then count each.

Test yourself

Can you spot these traps under time pressure?

Take a timed quiz on Permutations & Combinations and see if you avoid the mistakes above.