number of ways of rearranging the word mississippi?

Question

hello
i have got the number of ways to be 11! / 4!4!2! which equals 34650.
i need to find out the number of ways the result starts with a p and ends with an s. The answer says it is 9! / 3!4! but could you please tell me why.
Also is it possible to work out the solution using n choose r

thank you!

Answer 1

For mississippi
there are 11 characters
4 characters are s
4 characters are i
2 characters are p
so no of ways = 11!/ 4!4!2!

To start word with p and end with s
we can fix 1 p at start and 1 s at end.
So, only left 9 characters
4 charatcers are i
3 charaters are s
1 character is p
so no of way = 9! / 4!3!

Answer 2

Of course, the p and s has been selected and don't really count.......theses letters being fixed in their specific positions.
Only 9 letters, therefore, can be rearranged and as you know there would be 9! ways of doing this if all the letters were different; however, they're not all different. The 3 remaining "s" can be rearranged among them selves 3! ways without making any difference to the result of the search and, likewise, the 4 "i" can be rearranged 4! ways without affecting the solution of the problem.
The remaining "p" can be rearranged 1! ways.
So, for instance, if you had 9 different letters a,b,c,d,e,f,g,h and i you could rearrange them 9! ways. Yet, with the word mississippi and the two letter restriction, we need to divide the 9! by 3!, 4! and 1! [which is just one anyway]
The result is 2520.
There is no "n" or "r" in your specified word, so you can't rearrange letter you haven't got.
Let's try another word....."never". How many ways can we rearrange these letters? The answer is 5!/2!. Why? There are 5 ways of placing the first letter, 4 of placing the second...etc. But, because you have 2 "e" you, have to divide by 2. In the word "never" you could swap the second "e" and fourth "e" but still have "never" This would be true of every other word you made from these five letter. Hence the need to divide by 2.
These types of problems are very confusing.
I hope that helps a little.

Answer 3

Mond, once u fix two of the letters, these two letters each have only 1 choice each for their placings in the generated words, hence the 11 places that were originally created for p, say, and the remaining 10 for the s are now factored out of the numerator and replaced with 1's. With respect to the denominator, the 4!4!2! relates to the number of s's, i's, and p's in the word who have a free reign to appear wherever they like in the rearranged words, this is factorialised so to echo the complex number of possible ways involved in the rearrangement. This denominator will be reduced to (4-1)!4!(2-1)! since the p and s must now be fixed throughout the ensemble of newly constructed words, which means the number of ways becomes 9!/3!4!.....I hope thats helped you!!!

Answer 4

Your question is equivalent to ordering the letters missisipi (mississippi, without a p and without a s). This is because the p and s are fixed:

P_________S

The only letters you need to worry about rearranging are the ones int the middle.

Therefore, since there are 9 letters in the middle, 3 S's, and 4 I's, there are 9!/(3!*4!).

I don't think that you want to use nCr because you are talking about permutations (in which the order matters), not combinations (in which the order does not).

Answer 5

mississippi
There are 11 letters in this word, out of which 4 are similar ones of one type(s' s), 4 letters similar of the second type( i' s) ,2 are similar of the third type
( p' s) and one distinct m.
So your answer to the first part i.e. 11!/(4! *4! * 2!) is perfectly correct.
For the second part where we want p in the first place and s in the last place, we proceed as follows:
First fix one p in the first place and one s in the last place as desired. Now there are left 9 letters, out of which 3 s' s, 4 i' s, one p and one m. These can be arranged in 9 places in 9!/(3! *4! ) ways, as there are three letters similar of one type(s' s) and 4 letters similar of the second type( i ' s) and other letters are distinct.

Answer 6

Starts with a "p" you know you have two choices
Ends with an "s" you know you have four choices

I'd do this

(2) (4)
and fill in the missing letters
(2)(9!)(4)/ 4!4!2!

Answer 7

if you are talking about merely rearranging all or some of the letters you can make 28 diff words from the given letters