Permutation & Combination formulas
Master Permutation & Combination through 32 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.
Permutation & Combination, every formula
32 formulas, typeset and free. Print it, or keep it open beside your practice.
Fundamental principle (multiplication)
an event in m ways AND another in n ways
Q1NumericalMultiplication principleA person has $3$ shirts and $4$ trousers. The number of shirt-trouser combinations is:Fundamental principle (addition)
an event in m ways OR another in n ways
Q1NumericalAddition principleA student can travel by $3$ buses or $2$ trains. The number of ways to travel is:Factorial
n a non-negative integer
Q1NumericalFactorialThe value of $5!$ is:Permutations nPr
0 <= r <= n
Q1NumericalPermutations nPrThe value of ${}^{5}P_3$ is:Permutations of all n
Q1NumericalArrange allThe number of ways to arrange $4$ distinct books on a shelf is:Property nP0 and nP1
Q1MCQProperty of nPrThe value of ${}^{7}P_0$ is:- A$1$
- B$0$
- C$7$
- D$7!$
- A
Permutations with repetition allowed
r places, each filled by any of n things
Q1NumericalRepetition allowedThe number of $3$-digit strings formed from the digits $1,2,3,4,5$ with repetition allowed is:Permutations with alike objects
p, q, r alike of each kind, p+q+r=n
Q1NumericalWord with alike lettersThe number of distinct arrangements of the letters of the word $\text{LEVEL}$ is:Circular permutations of n things
clockwise and anticlockwise distinct
Q1NumericalCircular arrangementThe number of ways $5$ people can be seated around a round table is:Circular (necklace / garland)
clockwise and anticlockwise same
Q1NumericalNecklaceThe number of distinct necklaces using $4$ different beads is:Combinations nCr
0 <= r <= n
Q1NumericalCombinations nCrThe value of ${}^{6}C_2$ is:Property nC0, nCn, nC1
Q1MCQProperty nC0The value of ${}^{n}C_0$ is:- A$1$
- B$n$
- C$0$
- D$n!$
- A
Symmetry of nCr
Q1MCQSymmetry${}^{10}C_7$ equals:- A${}^{10}C_3$
- B${}^{10}C_7$ only
- C${}^{7}C_3$
- D${}^{10}C_{17}$
- A
Pascal's rule
Q1MCQPascal's rule${}^{n}C_r+{}^{n}C_{r-1}$ equals:- A${}^{\,n+1}C_r$
- B${}^{\,n+1}C_{r-1}$
- C${}^{n}C_{r+1}$
- D${}^{\,n-1}C_r$
- A
nCr = nCs condition
Q1NumericalnCr = nCsIf ${}^{15}C_{3r}={}^{15}C_{r+3}$ (with $3r\neq r+3$), then $r$ is:Ratio nCr / nC(r-1)
Q1MCQRatio identity$\dfrac{{}^{n}C_r}{{}^{n}C_{r-1}}$ equals:- A$\dfrac{n-r+1}{r}$
- B$\dfrac{n-r}{r}$
- C$\dfrac{r}{n-r+1}$
- D$\dfrac{n}{r}$
- A
Greatest binomial coefficient position
value of r making nCr largest
Q1NumericalGreatest coefficientFor $n=8$, the value of $r$ giving the greatest ${}^{8}C_r$ is:Total selections (at least one)
from n distinct things
Q1NumericalTotal selectionsThe number of ways to select at least one from $4$ distinct objects is:Selections from alike things
one or more, p alike of a kind etc.
Q1NumericalSelections from alikeThe number of ways to select one or more fruits from $3$ apples and $2$ oranges (fruits of a kind alike) is:Gap method (no two together)
choose r of n in a row, no two adjacent
Q1NumericalNo two togetherFrom $7$ chairs in a row, the number of ways to choose $3$ so that no two are adjacent is:Always-include / never-include (perm.)
one particular thing
Q1NumericalAlways includeThe number of arrangements of $4$ of $6$ distinct books such that a particular book is always included:Division into groups (unequal)
groups of sizes m, n, p
Q1NumericalUnequal groupsThe number of ways to divide $6$ people into groups of $1,2,3$ is:Division into equal groups
m groups of n each, order of groups not important
Q1NumericalEqual groupsThe number of ways to divide $4$ people into $2$ groups of $2$ each (groups unlabeled) is:Distribution to distinct persons
give m to one, n to another (both ways)
Q1NumericalDistribute to personsThe number of ways to give $4$ distinct books to two persons so one gets $2$ and the other gets $2$ is:Non-negative integer solutions
Q1NumericalNon-negative solutionsThe number of non-negative integer solutions of $x+y+z=5$ is:Positive integer solutions
each x_i >= 1
Q1NumericalPositive solutionsThe number of positive integer solutions of $x+y+z=6$ is:Derangements
no object in its own place
Q1NumericalDerangementsThe number of ways to place $4$ letters into $4$ addressed envelopes so that no letter is in its correct envelope is:Lines from n points (no 3 collinear)
straight lines joining n points
Q1NumericalLines from pointsThe number of straight lines through $6$ points (no three collinear) is:Triangles from n points
no three collinear
Q1NumericalTriangles from pointsThe number of triangles formed by $6$ points (no three collinear) is:Diagonals of an n-gon
Q1NumericalDiagonalsThe number of diagonals of a regular hexagon is:Exponent of prime p in n!
highest power of prime p dividing n!
Q1NumericalExponent of primeThe exponent of $5$ in $30!$ is:Number of divisors
N = p1^a1 p2^a2 ... pk^ak
Q1NumericalNumber of divisorsThe number of divisors of $360=2^{3}\cdot3^{2}\cdot5$ is:
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