Counting

In combinatorics there are two types of counting problems.

For example, in how many ways can balls of different colors be arranged ? Let us consider 3 balls of colors red , blue and green respectively.

Possible different arrangements of 3 balls are :

  • RGB
  • RBG
  • BRG
  • BGR
  • GBR
  • GRB

Total count is 6 and it is given by . Or for balls it's .

Counting number of selections (Combinations)

Let us count the number of ways of selecting items out of distinguishable items :

Counting number of arrangements (Permutations)

Let us count the number of ways of selecting items out of distinguishable items and then arranging them in some order.:

Methods for calculating :

int nCr(int n,int r){
    int res = 1 ;
    r = min(r,n-r) ; // nCr = nC(n-r) 
    for(int i=r;i>=1;i--){
        res = res * n ;
        res /= i ;
        n -- ; 
    }
    return res ;
}
// this code works properly for small size input

we can also use following recurrence to build up the dynamic programming for repeated queries.

 
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