Hi, Fahim,
Quote:
Originally Posted by fahim mohammed
Hmmm..Not necessarily...
Can 2 colors be repeated?

Perhaps you mean can
a color be repeated. If there were two colors that were repeated, there would be a minimum of four stripes altogether, certainly possible, but not found in the class of flag designs we have been looking at.
************
"Permutation" in the strict sense has a unique meaning, and does require nor admit of further qualification.
That sense operates upon a set of objects with no duplication (for example, the three objects A, B, and C). The "permutations" of that set are all the sequences (orders) in which those three objects can be placed:
ABC
ACB
BAC
BCA
CAB
CBA
A related matter, not strictly "permutations", operates upon a set with repetition, as for example the set A,A,B.
That set can be ordered in these distinct ways:
AAB
ABA
BAA
Then there is another "less strict" meaning of "permutation"!
This operates on a set of
n objects, all different, and considers all the possible ways we can draw
m of those and then, having drawn each such subset, all the different ways it can be ordered, the numerical result being the totality of all that.
For example, if we consider all the "permutations of three things, taken two at a time" (which is how that form of the notion is often expressed), where the three objects are A, B, and C, the permutations are:
AB
BA
AC
CA
BC
CA
This is in effect the notion of
combinations concatenated with the notion (in the strict sense) of
permutations.
That is, these are the combinations of three things taken two at a time (and here the sequence in which I write them is not significant):
A,B
A,C
B,C
But for each of those, there are two permutations (in the strict sense); that it, two ways they can be sequenced.
So from A,B we get AB and BA; from A,C we get AC and CA; from B,C we get BC and CB (overall, the six listed earlier).
There are of course many variations on these themes which are of importance in various real "problems" (including in the area of statistics), and often given descriptions involving the word "permutation", but strictly, only the first I showed is properly called the matter of "permutation".
Thanks.
Best regards,
Doug