Thread: Permutation of the word STATISTIC

Q: In how many ways can the letters in the word STATISTIC be arranged?

Now, I am not interested in the actual answer but rather the method whereby the problem is solved. We can begin by computing the number of arrangements while neglecting the repeated letters, which yields a total number of ways of arrangements.

Now here's is where I am confused: Why do we then followingly divide by the number of arrangements that can be made of the repeated letters? The right answer is obtainable by dividing by the (permutations of S), (permutations of T) and again (permutations of I).

Are we not supposed to take the overcounted arrangements and successively subtract the arrangements of the repeated letters, instead of dividing?

Originally Posted by MathCrusader

Q: In how many ways can the letters in the word STATISTIC be arranged?

Now, I am not interested in the actual answer but rather the method whereby the problem is solved. We can begin by computing the number of arrangements while neglecting the repeated letters, which yields a total number of ways of arrangements.

Now here's is where I am confused: Why do we then followingly divide by the number of arrangements that can be made of the repeated letters? The right answer is obtainable by dividing by the (permutations of S), (permutations of T) and again (permutations of I).

Are we not supposed to take the overcounted arrangements and successively subtract the arrangements of the repeated letters, instead of dividing?

The best way to see is to use a simple example.
Consider for our 'word'. Put subscripts in.
now we have six different letters that can be rearranged in ways.

Here are six, , of them:







Notice that the B's are in the same relative position, only the subscripts changed.
If we remove the subscripts then all six of those would be identical.

Thus the total is divided by to account for duplication.

Does that help.

Originally Posted by Plato

The best way to see is to use a simple example.
Consider for our 'word'. Put subscripts in.
now we have six different letters that can be rearranged in ways.

Here are six, , of them:







Notice that the B's are in the same relative position, only the subscripts changed.
If we remove the subscripts then all six of those would be identical.

Thus the total is divided by to account for duplication.

Does that help.

I understand your example, however I still can't we see division before subtraction is justified. We're taking a huge number 6!, i.e. the total number of arrangements, and dividing it by the arrangements of the B's i.e. 3!. We have 3! superfluous arrangements, so if one wishes to get rid of these extraneous arrangements from total number of arrangements, one would naturally want to subtract, but evidently no. Why?

Originally Posted by MathCrusader

I understand your example, however I still can't we see division before subtraction is justified. We're taking a huge number 6!, i.e. the total number of arrangements, and dividing it by the arrangements of the B's i.e. 3!. We have 3! superfluous arrangements, so if one wishes to get rid of these extraneous arrangements from total number of arrangements, one would naturally want to subtract, but evidently no. Why?

Don't you understand that there is no subtraction whatsoever in these problems.

The number of ways to rearrange the word is
. We divide to eliminate duplication.

There is no subtraction to it.

Originally Posted by Plato

Don't you understand that there is no subtraction whatsoever in these problems.

The number of ways to rearrange the word is
. We divide to eliminate duplication.

There is no subtraction to it.

I do understand that, but I am still asking why since I want to deepen my understanding of this concept.
There must be rationale behind dividing in order to reduce the extraneous arrangements, as opposed to subtracting the extraneous arrangements.

Originally Posted by mathcrusader

i am still asking why since i want to deepen my understanding of this concept. There must be rationale behind dividing in order to reduce the extraneous arrangements, as opposed to subtracting the extraneous arrangements.

That is eleven different letters. They can be rearranged in

In any one of those arrangements
The I's represent ways.
The S's represent ways.
The P's represent ways.

So we have over-countered by times.
So divide.