Quantitative Aptitude > Permutations and Combinations

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The number of arrangements of 'n' distinct object into 'n' distinct position is given by n,n-1,n-2,........3,2,1, which is often denoted as n! ( read as n 'factorial'). The number of arrangements is also know as “Permutations". Thus number of permutation of n distinct object into n distinct places is denoted as , read as permutations of n distinct objects taken n at a time.

 

The number of Permutations of n distinct objects taken from a group of 'r' distinct objects, in n distinct place is given by =

 

The number of Permutations of n object out of which m are of one type, q are of second type etc. Such that n ≠ m ≠q  …. is given by

 

The number of Permutation of n distinct objects taken all as times around a circle is given by (n-1)! And accordingly as clockwise and anticlockwise arrangements are treated as different or same. This formula is valid for a necklace.

 

Number of arrangements of n objects, such that any p out of those occur together is given by

(n-p+1) ! * p !.

 

Number of arrangements of n distinct things, when any object can be repeated any number of times is

 

given by .

 

 

Number of combinations of r distinct things taken out of n distinct things is given by

 

=

 

Please note in case of combinations also known as selections the order in which things are chosen is not important. Thus if we consider 4 distinct alphabets viz. A,B,C & D then no of permutations of any 3 alphabets out of is given by ABC

ABD

ACB

ADB

ACD

ADC, etc whereas no. of selections is ABC,ABD,ACD

and BCD i.e. 4 only.

 

Number of ways of selecting r things out of n distinct things is same as number of ways of selecting n-r things out of n things. i.e.

 

 

 

 

No of ways in which none or some objects can be chosen out of n distinct of objects is given by

 

 

 

The total number of ways in which n identical things can be distributed into r distinct boxes, such that one or more than one box may remain empty, but not all the boxes can be empty is given by

 

If blank boxes are not allowed i.e each box has at least one object then no of ways is given by

 

Number of non-negative solutions of is given by

 

Number of Non – negative solution of x ₁ +x ₂ …......... x _r = n is given by