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Original Question:

How many arrangements of the letters in MISSISSIPPI have no consecutive S's?

Solution:

$8\choose4$ * (7!/4!*2!)

Logical Question:

I really don't understand what the question is asking me to do. It is worded so confusing. Could you please clarify this question for me ? Also tell me how did you understand the question (the hints and tricky stuff).

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    What's confusing? It's asking you how many ways to arrange the letters MISSISSIPPI so that no two Ses are next to each other. SMSISISIPPI is one way, SMISISISPPI is another. – 2017-02-17
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    Oh I see, I just misunderstood a small thing :D You are right but the answer is not only limited to "SMSISISIPPI is one way, SMISISISPPI is another" to these two. I will respond to the question myself. – 2017-02-17
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    Of course those aren't the *only* two!!! There 7350 of them! – 2017-02-17
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    Haha I know they were just an example, don't take it serious. :) – 2017-02-17

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Simplified Solution:

You got 7!/4!2! for the arrangements of the 7 letters MIIIPPI without the letter S. And you have to choose arrangements that one S is not repeated after another, so no SS or SSS or SSSS.

$8\choose 4$ is used because we have 1 M 2 I 3 I 4 I 5 P 6 P 7 I 8 eight places to place our 4 S letters. The order of S does not matter which indicates use of combinations, so n = 8 and r = 4, then 8 choose 4.

Therefore, there are 7350 arrangements of non-consecutive S in MISSISSIPPI

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I think this really boils down to combinatorics, specifically arrangements. These are really cool questions, and I'm glad you're being asked questions like this in discrete mathematics because they're really interesting and help you hone your analytical skills.

Anyways, on to how to think about this question. It is classic and I'm sure you can find tons of examples of this specific problem being solved. I'm sure you can see that Mississippi (we can ignore case here) is one specific arrangement of $m i^4s^4p^2, miiiisssspp$, or even just $\{m,i,s,p\}$

So, choose any letter. Let's just go left to right. First you're going to fix a spot for the $m$, which can be done 11 ways (Mississippi is an 11 character word). Now, your $i$, you have 10 places left and 4 $i$'s to use. This is going to be $10\choose{4}$, however you're going to need to remove the choices in there that place $i$s together. Now do that same thing for for $6\choose4$ with $s$, and $2\choose2$ for your $p$s.

Hopefully that guides you to understanding the problem a little better. I know in your case specifically we're just talking about the $s$ being together, but Math Stackexchange is about helping you be a better mathematician, so now you have the basic idea of what the question is asking and you can apply the general "no letters touching" to "we only care if one letter is touching itself" $_{lol}$

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    thank you for your time to share your knowledge. I just want to add something to your helpful information, for 11 choose 11 ways, 10 choose 4 , 6 choose 4, and 2 choose 2, there must be rule of product used. The reason is that each stage like 6 choose 4 is dependent on the previous stage 10 choose 4. Thus, rule of product is used. Another approach would be to use combinations with repetitions: 11!/4!4!2! – 2017-02-17