Sunday, February 6, 2011

Important Concepts on Permutation

Permutation

Number of permutations of ‘n’ different things taken ‘r’ at a time is given by:-
nPr       =            n!/(n-r)!
Proof:     Say we have ‘n’ different things a1, a2……, an.
Clearly the first place can be filled up in ‘n’ ways. Number of things left after filling-up the first place = n-1
So the second-place can be filled-up in (n-1) ways. Now number of things left after filling-up the first and second places = n - 2
Now the third place can be filled-up in (n-2) ways.
Thus number of ways of filling-up first-place = n
Number of ways of filling-up second-place = n-1
Number of ways of filling-up third-place = n-2
Number of ways of filling-up r-th place = n – (r-1) = n-r+1
By multiplication – rule of counting, total no. of ways of filling up, first, second --  rth-place together :-
n (n-1) (n-2) ------------ (n-r+1)
Hence:
nPr       = n (n-1)(n-2) --------------(n-r+1)
= [n(n-1)(n-2)----------(n-r+1)] [(n-r)(n-r-1)-----3.2.1.] / [(n-r)(n-r-1)] ----3.2.1
nPr = n!/(n-r)!
Number of permutations of ‘n’ different things taken all at a time is given by:-
nPn               =         n!
Proof   :
Now we have ‘n’ objects, and n-places.
Number of ways of filling-up first-place  = n
Number of ways of filling-up second-place = n-1
Number of ways of filling-up third-place  = n-2
Number of ways of filling-up r-th place, i.e. last place =1
Number of ways of filling-up first, second, --- n th place
= n (n-1) (n-2) ------ 2.1.
nPn =  n!
Concept. 
We have   nPr  =     n!/n-r
Putting r = n, we have :-
nPr  =   n! / (n-r)
But   nP=  n!
Clearly it is possible, only when  n!  = 1
Hence it is proof that     0! = 1

Note : Factorial of negative-number is not defined. The expression  –3! has no meaning.

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