# Permutations and combinations

• B
The total number of different combinations of one or more letters which can be made from the letters of the word MISSISSIPPI is?

First i dont understand what the question means
and second my answer is completely different from that in my book

My working-
since there are 11 letter 4 I 4 S 2 P
Number of combinations= (11!) /(4! 4! 2!)
But this is not even close to the answer.

## Answers and Replies

phyzguy
It says, "of one or more letters". So you need to count all of the one letter combinations, the two letter combinations, the three letter combinations, and so on, then add them all up. You only counted the 11 letter combinations.

mfb
Mentor
I'm not sure how to interpret combinations here.
Are MISSISSIPPI and IIIIMPPSSSS different combinations, or are they different ways to describe the same combination "take all 11 letters"?
The first interpretation makes the problem very messy casework, the second one has a short solution.

I'm not sure how to interpret combinations here.
Are MISSISSIPPI and IIIIMPPSSSS different combinations, or are they different ways to describe the same combination "take all 11 letters"?
The first interpretation makes the problem very messy casework, the second one has a short solution.
the answer is 149

mfb
Mentor
The second interpretation then.
"Take all 11 letters" is one case. "Take all apart from the M" is another, and so on.

If you don't know how to start, it is always useful to look at smaller cases.

How many options are there for "AAA" or "AAAA"? What about "ABC"? "AAABBB"? Do you see some pattern?

Still can't understand anything :(

something like 1 letter combinations then 2 letter then so on?

mfb
Mentor
Yes.

As an example, for "AAA" there are the combinations "A", "AA" and "AAA". The combination "" doesn't have any letter so it doesn't count. What about the other examples I mentioned?

ok so like this
there are 2p 4iand 4s

2+4+4+1!+2!+3!+4! (cause 4 different letters)

=43 oops

PeroK
Homework Helper
Gold Member
2021 Award
Still can't understand anything :(

Given the eleven letters MISSISSIPPI can you make the following words?

RIVER
MIPS
PIPPI
X
IS

Some yes, some no. So, how many words can you make in total?

Ah, I see that's not the question! I guess permutations would be words.

Last edited:
mfb
Mentor
ok so like this
there are 2p 4iand 4s

2+4+4+1!+2!+3!+4! (cause 4 different letters)

=43 oops
See above: Start with easier cases to see how it works.

A Physics Enthusiast
The total number of different combinations of one or more letters which can be made from the letters of the word MISSISSIPPI is?

First i dont understand what the question means
and second my answer is completely different from that in my book

My working-
since there are 11 letter 4 I 4 S 2 P
Number of combinations= (11!) /(4! 4! 2!)
But this is not even close to the answer.
What is the answer in your book ?

What is the answer in your book ?
149

A Physics Enthusiast
The total number of different combinations of one or more letters which can be made from the letters of the word MISSISSIPPI is?

First i dont understand what the question means
and second my answer is completely different from that in my book

My working-
since there are 11 letter 4 I 4 S 2 P
Number of combinations= (11!) /(4! 4! 2!)
But this is not even close to the answer.
OK. I've got the answer.
We have the multiset
$$\left\{\text{M}^1, \, \text{I}^4,\, \text{S}^4,\, \text{P}^2\right\}\tag*{}$$
Now, we can form the combination generating function by assuming each letter to be x.
$$(1+x)(1+x+x^2+x^3+x^4)^2(1+x+x^2)$$
$$=x^{11}+4x^{10}+9x^9+15x^8+21x^7+25x^6+25x^5+21x^4+15x^3+9x^2+4x+1$$
Now, add the coefficients except the constant term. You'll get 149.
Read this if you want to clear your understanding.

Janosh89
mfb
Mentor
@T13091999: While that is not impossible, it is like calculating 4+7 with Fourier transformations. It is way more complicated than necessary.
In addition, it is OP's homework, not yours.

A Physics Enthusiast
@T13091999: While that is not impossible, it is like calculating 4+7 with Fourier transformations. It is way more complicated than necessary.
In addition, it is OP's homework, not yours.
I was just trying to help. I got the answer. Still, I am not happy with my way of doing it. Do you have any sugesstion ?
P.S. : I did the sum as I was trying to help the OP.

I liked the question, has this been posted here from another forum in order to engage more people?

mfb
Mentor
Do you have any sugesstion ?
Post #5 should help enough to find a short, easy, general formula. With the right approach can calculate the answer in your head in a few seconds.

PeroK
Homework Helper
Gold Member
2021 Award
Still, I am not happy with my way of doing it. Do you have any sugesstion ?

Think of it like this. Someone has four bags, and each bag contains tiles with one letter on them. One is a bag of M's, another a bag of I's etc. (For the general problem you can imagine 26 bags each containing tiles with one letter.)

You want to make a combination of some or all of the letters in Mississippi. You need to get your letters from the person with the bags of letters. What do you ask for?

A Physics Enthusiast
Post #5 should help enough to find a short, easy, general formula. With the right approach can calculate the answer in your head in a few seconds.
Seems like I killed the mosquito with my machine gun.
$$\Large \ddot \smile$$

I actually liked what came at the end of "a hail of bullets". Something to get to grips with before senescence takes its ugly grip.