PERMUTATIONS & COMBINATIONS
COMBINATIONS
Number of selections of n different objects taken ‘r‘ at a time.
If the order doesn't matter, it is a Combination.
Number of combinations of n dissimilar things taken 'r' at a time is denoted by nCr & is given by nCr =n!/(n-r)!r!
Number of combinations of n different things taken ‘r’ at a time in which ‘p’ particular things will always occur is (n–p)C(r–p)
Number of combinations of n dissimilar things taken 'r' at a time in which 'p' particular things will never occur is (n–p)Cr
PERMUTATIONS
Number of arrangements of n different objects taken ‘r’ at a time.
If the order does matter it is a Permutation.
A Permutation is an ordered Combination.
Permutations of n different things taken 'r' at a time is denoted by nPr
The total number of arrangements of n things taken ‘r’ at a time, in which a particular thing always occurs = (n–1)P(r–1).
The total number of permutations of n different things taken ‘r’ at a time in which a particular thing never occurs = (n–1)Pr.
The number of arrangements when p of them are of one kind, q of another kind, r is still of another kind and so on, the total number of permutations is given by
n!/(p! q! r !.......)
Number of circular permutations of n things all taken at a time = (n – 1)!
Number of circular permutations of n different things taking ‘r’ at a time =nPr/r
Other Results:
nC0=1
nCn=1
nCr=nCn-r
nCr+nC(r-1)=(n+1)Cr