EXAMPLES for Combinatorics
------------------------------

   COMBINATION
  -------------
values: n,k :
   = sum of groups consisting of k-number of elements in
	 a set of n-number of elements
where the order of elements is not considered ;
	each element in a group is unique, does not recur

Formula:
		n!
   C(n,k) = -----------
	     (n-k)!k!

Examples:

1. elements A,B,C (n=3) in couples (k=2)
n=3, k=2, C(n,k)=3
	AB
	BC
	AC

2. elements A,B,C,D (n=4) in trios (k=3)
n=4, k=3, C(n,k)=4
	ABC
	ABD
	BCA
	BCD

3. elements A,B,C,D,E (n=5) in groups of 5 (k=5)
n=5, k=5, C(n,k)=1
	ABCDE

There are no other possible combinations because the order of elements
in a group is neglected, so these listings are complete ;
the combinations of AB and BA are identical,
so expressed only once as AB.


   VARIATION
  -----------
values: n,k :
   = sum of groups consisting of k-number of elements in
	 a set of n-number of elements
where the order of elements is considered ;
	each element in a group is unique, does not recur

Formula:
		n!
   V(n,k) = -----------
	      (n-k)!

Examples:

1. elements A,B,C (n=3) in couples (k=2)
n=3, k=2, V(n,k)=6
	AB	BA
	BC	CB
	AC	CA	

2. elements A,B,C,D (n=4) in trios (k=3)
n=4, k=3, V(n,k)=24
	ABC	ACB	ABD	ADB	ACD	ADC
	BAC	BCA	BAD	BDA	BCD	BDC
	CAB	CBA	CAD	CDA	CBD	CDB
	DAB	DBA	DAC	DCA	DBC	DCB

3. elements A,B,C,D (n=4) in couples (k=2)
n=4, k=2, V(n,k)=12
	AB	AC	AD
	BA	BC	BD
	CA	CB	CD
	DA	DB	DC

Notice that the order of elements in a group is considered ;
the variations of AB and BA are not identical.


   PERMUTATION
  -------------
value: n :
   = sum of groups consisting of n-number of elements in
	 a set of n-number of elements
where the order of elements is considered ;
     each element in a group is unique, does not recur ;
= equivalent to operation of variation if n=k

Formula:
	
   P(n) = n!

Examples:

1. elements A,B (n=2)
n=2, P(n)=2  (the same as )

2. elements  A,B,C (n=3)
n=3, P(n)=6  (the same as variations where n=3, k=3)
	ABC	ACB
	BAC	BCA
	CAB	CBA

3. elements A,B,C,D (n=4)
n=4, P(n)=24  (the same as variations where n=4, k=4)
	ABCD	ABDC	ACBD	ACDB	ADBC	ADCB
	BACD	BADC	BCAD	BCDA	BDAC	BDCA
	CABD	CADB	CBAD	CBDA	CDAB	CDBA
	DABC	DACB	DBAC	DBCA	DCAB	DCBA


   VARIATION WITH RECURRENCE
  ---------------------------
values: n,k :
   = sum of groups consisting of k-number of elements in
	 a set of n-number of elements
where the order of elements is considered ; 
	elements can recur in a group

Formula:		
	      k
   Vr(n,k) = n
		(k-power of n)

Examples:

1. elements A,B (n=2) in couples (k=2)
n=2, k=2, Vr(n,k)=4
	AA	AB
	BA	BB

2. elements A,B,C (n=3) in couples (k=2)
n=3, k=2, Vr(n,k)=9
	AA	AB	AC
	BA	BB	BC
	CA	CB	CC

3. elements A,B,C (n=3) in trios (k=3)
n=3, k=3, Vr(n,k)=27
	AAA	AAB	AAC
	ABA	ABB	ABC
	ACA	ACB	ACC
	BAA	BAB	BAC
	BBA	BBB	BBC
	BCA	BCB	BCC
	CAA	CAB	CAC
	CBA	CBB	CBC
	CCA	CCB	CCC


   PERMUTATION WITH RECURRENCE
  ---------------------------
values: n,r :
	= sum of groups consisting of n-number of elements in
	 a set of n-number of elements
where the order of elements is considered ;
     each element in a group is unique, does not recur ;
     elements in the total set (not in a group) can occur r-times,
     it means r-number of elements are identical

Formula:		
	      n!
   Pr(n,r) = ----
     	      r!

Examples:

1. elements A,A (n=2) - two are identical (r=2)
n=2, r=2, Pr(n,r)=1
	AA

2. elements A,A,B (n=3) - two are identical (r=2)
n=3, r=2, Pr(n,r)=3
	AAB	ABA	BAA

3. elements A,A,A,B (n=4) - three are identical (r=3)
n=4, r=3, Pr(n,r)=4
	AAAB	AABA	
	ABAA	BAAA


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