34 days to CAT

Lesson 16 min· Section 1 of 4

The straight-up lesson, built for the exam.

Permutation & Combination

Formula sheet

Everything for this chapter, in one place.

The two tools


Order matters — arrangements	$^nP_r = \dfrac{n!}{(n-r)!}$
Order does not — selections	$^nC_r = \dfrac{n!}{r!\,(n-r)!}$
Symmetry	$^nC_r = {}^nC_{n-r}$
Edges	$^nC_0 = {}^nC_n = 1$
Link	$^nP_r = {}^nC_r \times r!$

When some objects are identical


Arranging $n$ with repeats $p, q, \dots$	$\dfrac{n!}{p!\,q!\cdots}$
Round a circle	$(n-1)!$
Any subset of $n$ distinct things	$2^n$

Identical objects into distinct boxes — stars and bars


$n$ identical into $r$ boxes, empties allowed	$^{n+r-1}C_{r-1}$
Each box at least one	$^{n-1}C_{r-1}$
Each box at least $k_i$	hand out the minimums first, then use the "empties allowed" form
An upper limit as well	count without it, then subtract the violations

Counting the complement


"At least one …"	total $-$ "none"
"At least one digit repeated"	total $-$ "all digits distinct"
Onto functions, $m$ elements to $n$	$n^m - {}^nC_1(n-1)^m + {}^nC_2(n-2)^m - \cdots$

Numbers from digits


Sum of all numbers from $n$ distinct digits, each used once	$(n-1)! \times (\text{sum of digits}) \times \underbrace{11\cdots1}_{n}$
Each digit appears in each place	$(n-1)!$ times
Leading zero	count freely, then subtract the arrangements that start with 0
Divisible by 3	the digit sum must be divisible by 3
Divisible by 6	digit sum divisible by 3 and last digit even

Paths and pairs


Lattice paths, $m$ right and $n$ up	$^{m+n}C_m$
Via an intermediate point	multiply the two legs
Handshakes among $n$	$^nC_2$
Pigeonhole	with $n$ items in $k$ boxes some box holds at least $\left\lceil \frac{n}{k} \right\rceil$

Why this chapter matters

0.8

questions in a typical Quant section

18th

of 25 chapters

≈ 2 of 66 marks a section ride on it.

Asked in19 of 24 papers

20172025

Year by yearper section

2017: 1.62017: 1.6 per section
2018: 0.72018: 0.7 per section
2019: 0.32019: 0.3 per section
2020: 0.82020: 0.8 per section
2021: 12021: 1 per section
2022: 12022: 1 per section
2023: 0.32023: 0.3 per section
2024: 1.32024: 1.3 per section
2025: 0.32025: 0.3 per section

Question type23 asked

MCQ 35%Type-in 65%

Type-in answers carry no negative marking.

From every Permutation & Combination question in CAT papers since 2017.