Concept: Permutation & Combination
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CONTENTS
- Introduction
- Basic Principles of Counting
- Formulae
- Arrangements
- Selection
- Distribution of Objects in Groups of Defined Sizes
- Distribution of Similar Objects in Different Groups
- Geometry Based
Permutation means arrangement while Combination means selection
If A or B (exactly 1 of A or B) are two tasks that must be performed such that A can be performed in 'p' ways and B can be performed in 'q' ways, then A or B can be performed in p + q ways
If A and B are two tasks that must be performed such that A can be performed in 'p' ways and for each possible way of performing A, say there are 'q' ways of performing B, then the two tasks A and B can be performed in p × q ways
Number of ways arranging r people out of n in a row = n × (n-1) × (n-2) ... × (n-r+1) = nPr
nPr =
Number of ways selecting r people out of n = (No of ways of arrangement) ÷ r! = (n × (n-1) × (n-2) ... × (n-r+1)) ÷ r! = nCr
nCr =
Number of ways of arranging 'n' distinct items in a line is given by
n! ways
Number of ways of arranging 'n' items out of which `p' are alike, 'q' are alike, 'r' are alike in a line is given by
ways
Number of ways of arranging 'n' distinct items around a circle is given by (n - 1)! ways
Number of ways of arranging 'n' distinct items around a circle is given by ways
Out of 'p' distinct objects of type I, 'q' distinct objects of type II and 'r' distinct objects of type III, number of ways of selecting
any number of objects = 2p × 2q × 2r = 2p+q+r
at least one object = 2p+q+r - 1
at least one object of each type = (2p - 1) × (2q - 1) × (2r - 1)
Out of 'p' similar objects of type I, 'q' similar objects of type II and 'r' similar objects of type III, number of ways of selecting
any number of objects = (p + 1) × (q + 1) × (r + 1)
at least one object = (p + 1) × (q + 1) × (r + 1) - 1
at least one object of each type = p × q × r
The number of ways of dividing (p + q + r) distinct items into three groups containing p, q and r items respectively is (p ≠ q ≠ r) =
The number of ways of dividing 2p distinct items into two equal groups of p each
- when the two groups have distinct identity, is
- when the two groups don't have distinct identity, is
The number of ways of dividing 3p distinct items into three equal groups of p each
- when the three groups have distinct identity, is
- when the three groups don't have distinct identity, is
No of ways of distributing 'n' similar objects in r distinct groups = n+r-1Cr-1
No of ways of distributing 'n' similar objects in r distinct groups such that each group gets at least one object = n-1Cr-1
Number of terms in the expansion of (a + b)n = n+1C1 = n + 1
Number of terms in the expansion of (a + b + c)n = n+2C2
Number of terms in the expansion of (a + b + c + d)n = n+3C3
- Number of ways of selecting a square from a n × n square matrix is:
- Number of ways of selecting a rectangle from a m × n matrix is:
- Number of ways of traveling from lower left corner to upper right corner of a m × n square matrix when you can travel only in east and north direction
12 + 22 + 32 + ... + n2
mC2 × nC2
Number of unique lines that can be formed by joining any 2 points out of ‘p’ points of which ‘c’ points are collinear (c < p) = pC2 - cC2 + 1
Number of unique triangles that can be formed with three vertices from ‘p’ points out of which ‘c’ points are collinear (c < p) = pC3 - cC3