Permutation & Combination formulas

Every formula JEE Advanced expects you to know for Permutation & Combination, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 32
Subject: Mathematics
Exam: JEE Advanced
No sign-in needed: ₹0

Permutation & Combination, every formula

32 formulas, typeset and free. Print it, or keep it open beside your practice.

Fundamental principle (multiplication)
$ways = m \times n$
an event in m ways AND another in n ways
Fundamental principle (addition)
$ways = m + n$
an event in m ways OR another in n ways
Factorial
$n! = 1 \cdot 2 \cdot 3 \dots n = n (n - 1)!, 0! = 1$
n a non-negative integer
Permutations nPr
$^{n} P_{r} = \frac{n !}{( n - r )!} = n (n - 1) \dots (n - r + 1)$
0 <= r <= n
Permutations of all n
$^{n} P_{n} = n!$
Property nP0 and nP1
$^{n} P_{0} = 1,^{n} P_{1} = n$
Permutations with repetition allowed
$n^{r}$
r places, each filled by any of n things
Permutations with alike objects
$\frac{n !}{p ! q ! r !}$
p, q, r alike of each kind, p+q+r=n
Circular permutations of n things
$(n - 1)!$
clockwise and anticlockwise distinct
Circular (necklace / garland)
$\frac{( n - 1 )!}{2}$
clockwise and anticlockwise same
Combinations nCr
$^{n} C_{r} = \frac{n !}{r ! ( n - r )!} = \frac{^{n} P _{r}}{r !}$
0 <= r <= n
Property nC0, nCn, nC1
$^{n} C_{0} =^{n} C_{n} = 1,^{n} C_{1} = n$
Symmetry of nCr
$^{n} C_{r} =^{n} C_{n - r}$
Pascal's rule
$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$
nCr = nCs condition
$^{n} C_{r} =^{n} C_{s} \Rightarrow r = s or r + s = n$
Ratio nCr / nC(r-1)
$\frac{^{n} C _{r}}{^{n} C _{r - 1}} = \frac{n - r + 1}{r}$
Greatest binomial coefficient position
$r = \frac{n}{2} (n even), r = \frac{n \pm 1}{2} (n odd)$
value of r making nCr largest
Total selections (at least one)
$^{n} C_{1} +^{n} C_{2} + \dots +^{n} C_{n} = 2^{n} - 1$
from n distinct things
Selections from alike things
$(p + 1) (q + 1) (r + 1) \dots - 1$
one or more, p alike of a kind etc.
Gap method (no two together)
$^{n - r + 1} C_{r}$
choose r of n in a row, no two adjacent
Always-include / never-include (perm.)
$include: r \cdot^{n - 1} P_{r - 1}; exclude:^{n - 1} P_{r}$
one particular thing
Division into groups (unequal)
$\frac{( m + n + p )!}{m ! n ! p !}$
groups of sizes m, n, p
Division into equal groups
$\frac{( mn )!}{( n ! ) ^{m} m !}$
m groups of n each, order of groups not important
Distribution to distinct persons
$\frac{( m + n )!}{m ! n !} \times 2!$
give m to one, n to another (both ways)
Non-negative integer solutions
$x_{1} + x_{2} + \dots + x_{r} = n \Rightarrow^{n + r - 1} C_{r - 1}$
Positive integer solutions
$x_{1} + x_{2} + \dots + x_{r} = n \Rightarrow^{n - 1} C_{r - 1}$
each x_i >= 1
Derangements
$D_{n} = n! (1 - \frac{1}{1 !} + \frac{1}{2 !} - \dots + (- 1)^{n} \frac{1}{n !})$
no object in its own place
Lines from n points (no 3 collinear)
$^{n} C_{2}$
straight lines joining n points
Triangles from n points
$^{n} C_{3}$
no three collinear
Diagonals of an n-gon
$^{n} C_{2} - n = \frac{n ( n - 3 )}{2}$
Exponent of prime p in n!
$⌊ \frac{n}{p} ⌋ + ⌊ \frac{n}{p ^{2}} ⌋ + ⌊ \frac{n}{p ^{3}} ⌋ + \dots$
highest power of prime p dividing n!
Number of divisors
$(α_{1} + 1) (α_{2} + 1) \dots (α_{k} + 1)$
N = p1^a1 p2^a2 ... pk^ak

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Permutation & Combination formulas

Permutation & Combination, every formula

Fundamental principle (multiplication)

Fundamental principle (addition)

Factorial

Permutations nPr

Permutations of all n

Property nP0 and nP1

Permutations with repetition allowed

Permutations with alike objects

Circular permutations of n things

Circular (necklace / garland)

Combinations nCr

Property nC0, nCn, nC1

Symmetry of nCr

Pascal's rule

nCr = nCs condition

Ratio nCr / nC(r-1)

Greatest binomial coefficient position

Total selections (at least one)

Selections from alike things

Gap method (no two together)

Always-include / never-include (perm.)

Division into groups (unequal)

Division into equal groups

Distribution to distinct persons

Non-negative integer solutions

Positive integer solutions

Derangements

Lines from n points (no 3 collinear)

Triangles from n points

Diagonals of an n-gon

Exponent of prime p in n!

Number of divisors

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