MHB Permutation of Letters: 10 Choose 4 with 4 Letter Gap between P and S

[juantheron]

Total no. of permutation of the words $\bf{PERMUTATIONS}$ in which there are

exactly $4$ letters between $P$ and $S$, is

My TRY:: If we fixed $P$ and $S$, Then there are $10$ letters $\bf{ERMUTATIONS}$

out of $10$, we have to select $4$ letters of $4$ gap b/w $P$ and $S$ and then arrange in this Gap.

Is I am Thinking Right or Not.

If Not please explain me ,Thanks

 

[MarkFL]

Re: permutations

I would begin by observing we have 12 "slots" to fill. How many ways can the P and S be arranged, with 4 slots between them? Consider both orderings of these two letters.

 

[Opalg]

Total no. of permutation of the words $\bf{PERMUTATIONS}$ in which there are

exactly $4$ letters between $P$ and $S$, is

My TRY:: If we fixed $P$ and $S$, Then there are $10$ letters $\bf{ERMUTATION}\color{red}{\bf{S}}$ (Don't need that S there!)

out of $10$, we have to select $4$ letters of $4$ gap b/w $P$ and $S$ and then arrange in this Gap.

Is I am Thinking Right or Not.

Maybe the easiest way is to start by saying that there are $10!/2$ ways of ordering those ten letters other than the P and the S (the division by 2 is because the two Ts are indistinguishable, so interchanging them does not lead to anything new).

Now think about how many ways there are to insert the P and the S into the list. If the P comes before the S, then there are 7 ways to insert them, namely
P****S******
*P****S*****
**P****S****
***P****S***
****P****S**
*****P****S*
******P****S,
and there will also be 7 arrangements with the S before the P.