Permutation and combination formulas





Permutation and combination formulas

 

Permutation is a arrangement of objects or symbols in distinguishable sequences. Combination can be define as a selection of some or all of the number of different  objects. Combination is a un-ordered collection of unique sizes.

 

Some of the important formulas of permutation and combination are listed below:-

 

[Note: factorial n = the product of first ‘n’ natural numbers it is denoted by n! ]

 

i.e. n! = n(n – 1) (n – 2) …………. 3.2.1. Ex. 5! = 5 * 4 * 3 * 2 * 1 = 120 0! = 1

 
a. Number of permutations of ‘n’ different things taken ‘r’ at a time is given by:
P (n, r) = \left(\dfrac{n!}{(n- r)!} \right)

 
b. Number of permutations of n things, taken ‘r’ at a time when each things be repeated r-times is given by =n’.

 
c. Number n-thing taken all at a time, in which ‘p’ are of one type, ‘q’ of them are of second type, ‘r’ of them are of third type, and rest are all different is given by = \left(\dfrac{n!}{p!p!r!} \right)

 
d. There are two case of circular permutations:

 
i. If clockwise and anti-clock-wise orders are different, and then total number of circular-permutations is given by: (n – 1)!

 
ii. if clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by: \left(\dfrac{(n- 1)!}{2!}\right)

 
e. Total number of combinations of n object taken r at a time is denoted by C(n, r) and defined by C (n, r) = \left(\dfrac{n!}{(n- r)!r! }\right)

 
[Note: C (n, r) = C (n, n – r)]

 
f. Restricted – combinations

i. Number of combinations of ‘n’ different things taken ‘r’ at a time, when ‘p’ particular things are always included = C (n-p, r – p)

 
ii. Number of combinations of ‘n’ different things, taken ‘r’ at a time, when ‘p’ particular things to be excluded = C (n – p,  r)



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