PERMUTATION AND COMBINATION
Permutations :
The ways of arranging or selecting smaller or equal number of persons or objects from a group of persons or collection of objects with due regard being paid to the order of arrangement or selection are called permutations.
Combinations :
The number of ways in which smaller or equal number of things are arranged or selected from a collection of things where the order of selection or arrangement is not important are called combinations.
Difference between Permutations and Combinations
|
Permutations Selection is made. Beyond selection, order or arrangement is important. |
Combinations Selection is made. But arrangement or order is not important |
Formulas
Permutations :
nPr = n!/(n-r)!
Combinations :
nCr = n!/r!(n-r)!
Circular Permutations :
Case (i) :
Both clockwise and anti clockwise rotations are considered. (Hint : Every person has the same two neighbors) Then, the formula for circular permutations is
(n-1)!
Case (ii) :
Either clockwise or anti clockwise rotation is considered, not both. (Hint: No person has the same two neighbors) Then, the formula for circular permutations is
(n-1)! / 2
Shortcuts
1. nPr = n(n-1)(n-2)....to "r" terms.
Example : 7P3 = 7x6x5 = 210
2. nCr = [n(n-1)(n-2)...to "r" terms]/r!
Example : 7P3 = [7x6x5]/[3x2x1] = 35
3. nCr = nCn-r
(we will use this property only when we want to reduce the value of "r")
Example : 25P22 = 25P3
4. nPr = r! ⋅ nCr
5. nP1 = n
6. nC1 = n
7. nP0 = 1
8. nC0 = 1
9. nPn = n!
(No. of permutations of n things taken all at a time)
10. nCn = 1
(Explanation : nCn = nCn-n = nC0 = 1)
11. No. of Permutations of n things taken all at a time, when two particular things always come together is
= (n-1)!.2!
12. No. of Permutations of n things taken all at a time, when two particular things always do not come together is
= n!-(n-1)!.2!
13. The value of 0! = 1
Fundamental Principles of Counting
Multiplication rule :
There are two things, one can be done in "m" number of ways and the second can be done in "n" number of ways, Then the total number of ways of doing both the things is
= m x n
This rule is called multiplication rule.
AND ===> Multiplication
Addition rule :
There are two things, one can be done in "m" number of ways and the second can be done in "n" number of ways, Then the total number of ways of doing either the first one or second one, not both is
= m + n
This rule is called addition rule.
OR ===> Addition
If you need practice problems on permutations and combinations,
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WORD PROBLEMS
Word problems on simple equations
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Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
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Complementary and supplementary angles word problems
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Word problems on sets and venn diagrams
Pythagorean theorem word problems
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OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
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