Permutations
Consider a set of 4
objects. Now, suppose that we want to fill three positions with objects selected from the 4. We need to consider the order of the items in each set as well, then the number of possible ordered arrangements is 24 and they are
a b c
|
b a c
|
c a b
|
Dab
|
a b d
|
b a d
|
c a d
|
d a c
|
a c b
|
b c a
|
c b a
|
d b c
|
a c d
|
b c d
|
c b d
|
d b a
|
a d c
|
b d a
|
c d b
|
d c a
|
a d b
|
b d c
|
c d a
|
d c b
|
The number of
possible ordered arrangements can be computed as follows:
Since there are 3 positions and 4 objects, the
first position can be filled in 4 different ways. Once the first position is filled
the remaining 2 positions can be filled from the remaining 3 objects. Thus, the
second position can be filled in 3 ways. The third position can be filled in 2
ways. Then the total number of ways 3 positions can be filled out of 4 objects
is given by
(4)(3)(2) = 24.
In HANA, we can write the following sql to achieve the same result set:
do begin
t_data =
select 'a' as id
from dummy
union all
select 'b' as id
from dummy
union all
select 'c' as id
from dummy
union all
select 'd' as id
from dummy
;
select o.id, t.id, r.id, f.id
from :t_data as o
cross join :t_data as t
cross join :t_data as r
cross join :t_data as f
where o.id != t.id
and o.id != r.id
and o.id != f.id
and t.id != r.id
and t.id != f.id
and r.id != f.id
;
end;
;ID;ID;ID;ID
1 ;a ;b ;c ;d
2 ;a ;b ;d ;c
3 ;a ;c ;b ;d
4 ;a ;c ;d ;b
5 ;a ;d ;b ;c
6 ;a ;d ;c ;b
7 ;b ;a ;c ;d
8 ;b ;a ;d ;c
9 ;b ;c ;a ;d
10;b ;c ;d ;a
11;b ;d ;a ;c
12;b ;d ;c ;a
13;c ;a ;b ;d
14;c ;a ;d ;b
15;c ;b ;a ;d
16;c ;b ;d ;a
17;c ;d ;a ;b
18;c ;d ;b ;a
19;d ;a ;b ;c
20;d ;a ;c ;b
21;d ;b ;a ;c
22;d ;b ;c ;a
23;d ;c ;a ;b
24;d ;c ;b ;a
Combinations
In permutation,
order is important. But in many problems, the order of selection is not
important and interest centers only on the set of r objects. Let c denote the
number of subsets of size r that can be selected from n different objects.
How many committees
of two chemists and one physicist can be formed from 4 chemists and 3
physicists?
do begin
t_phys =
select 'P1' as id
from dummy
union all
select
'P2' as id
from dummy
union all
select
'P3' as id
from dummy
;
t_chem =
select 'C1' as id
from dummy
union all
select
'C2' as id
from dummy
union all
select
'C3' as id
from dummy
union all
select
'C4' as id
from dummy
;
select p.id, ca.id,
cb.id
from :t_phys as p
cross join :t_chem
as ca
cross join :t_chem
as cb
where ca.id != cb.id
and cb.id > ca.id
order by p.id,
ca.id, cb.id
;
;ID;ID;ID
1 ;P1;C1;C2
2 ;P1;C1;C3
3 ;P1;C1;C4
4 ;P1;C2;C3
5 ;P1;C2;C4
6 ;P1;C3;C4
7 ;P2;C1;C2
8 ;P2;C1;C3
9 ;P2;C1;C4
10;P2;C2;C3
11;P2;C2;C4
12;P2;C3;C4
13;P3;C1;C2
14;P3;C1;C3
15;P3;C1;C4
16;P3;C2;C3
17;P3;C2;C4
18;P3;C3;C4
