# Permutation of the word STATISTIC

• January 29th 2013, 09:20 AM
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 $9!$ 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 $2!$ (permutations of S), $3!$ (permutations of T) and again $2!$ (permutations of I).

Are we not supposed to take the overcounted arrangements and successively subtract the arrangements of the repeated letters, instead of dividing?
• January 29th 2013, 09:49 AM
Plato
Re: Permutation of the word STATISTIC
Quote:

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 $9!$ 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 $2!$ (permutations of S), $3!$ (permutations of T) and again $2!$ (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 $ABBBCD$ for our 'word'. Put subscripts in.
$AB_1B_2B_3CD$ now we have six different letters that can be rearranged in $6!$ ways.

Here are six, $3!$, of them:
$B_1ACB_3DB_2$
$B_1ACB_2DB_3$
$B_2ACB_1DB_3$
$B_2ACB_3DB_1$
$B_3ACB_2DB_1$
$B_3ACB_1DB_2$

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 $3!$ to account for duplication.

Does that help.
• January 29th 2013, 02:58 PM
Re: Permutation of the word STATISTIC
Quote:

Originally Posted by Plato
The best way to see is to use a simple example.
Consider $ABBBCD$ for our 'word'. Put subscripts in.
$AB_1B_2B_3CD$ now we have six different letters that can be rearranged in $6!$ ways.

Here are six, $3!$, of them:
$B_1ACB_3DB_2$
$B_1ACB_2DB_3$
$B_2ACB_1DB_3$
$B_2ACB_3DB_1$
$B_3ACB_2DB_1$
$B_3ACB_1DB_2$

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 $3!$ 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?
• January 29th 2013, 03:14 PM
Plato
Re: Permutation of the word STATISTIC
Quote:

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 $MISSISSIPPI$ is
$\frac{11!}{4!\cdot 4!\cdot 2!}$. We divide to eliminate duplication.

There is no subtraction to it.
• January 29th 2013, 03:35 PM
Re: Permutation of the word STATISTIC
Quote:

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 $MISSISSIPPI$ is
$\frac{11!}{4!\cdot 4!\cdot 2!}$. 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.
• January 29th 2013, 04:04 PM
Plato
Re: Permutation of the word STATISTIC
Quote:

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.

$MI_1S_1S_2I_2S_3S_4I_3P_1P_2I_4$
That is eleven different letters. They can be rearranged in $11!$

In any one of those arrangements
The I's represent $4!$ ways.
The S's represent $4!$ ways.
The P's represent $2!$ ways.

So we have over-countered by $4!\cdot 4!\cdot 2!$ times.
So divide.