Combinatorics - Permutations of abcdefg. Which is right?

[Alche]

Homework Statement

How many permutations of the letters a, b, c, d, e, f, g have either two or three letters between a and b? b _ _ a is also very much possible.

Homework Equations

ⁿP_r= n!/(n - r)!, where n >= r

The Attempt at a Solution

For this question there can be 4 cases which are as follows

1)when there are 4 letter words,

a _ _ b

from among 5 remaining letters 2 can be taken in 5P2 ways which can be arranged themselves in 2! ways and a and b can also be arranged among themselves in 2 ways, so

5P2*2!*2! = 80

2)5 letter words, here can be 3 cases too which are as follow:-

A) a _ _ b _
B) a _ _ _ b
c) _ a _ _ b

letters can be arranged here as

(5P3*3!*2)*3 = 2160.

3) when 6 lettered words are formed

a) a _ _ _ b _
b)a _ _ b _ _
c)_ a _ _ b _
d)_ a _ _ _ b
e) _ _ a _ _ b

here the letters can be arranged as

(5P4*4!*2)*5 = 28800

4)when 7 lettered word is formed

a) a _ _ _ b _ _
b) a _ _ b _ _ _
c) _ _ a _ _ _ b
d) _ _ _ a _ _ b
e) _ a _ _ b _ _
f) _ a _ _ _ b _
g) _ _ a _ _ b _

(5P5*5!*2)*7 = 201600

So now am getting the answer as 232640.

Please tell me am I right? If not then where am I making mistake.

 

[Outlined]

I think you are wrong as there are no 4 letter words. All your words are 7 letters (we are talking about permutations after all). The question is: how many do we have with your restriction. When a and b are in the following place

a _ _ b _ _ _

we have 5! words (as there are 5 places left for the remaining letters). However a and b can be placed in more ways. Count them too. Your answer at 4) is in the good direction.

my answer

Spoiler

1680

 

[Alche]

Thanks for replying.

I got it.

 

[HallsofIvy]

Apparently Alche is NOT talking about "permutations" but about both permutations and combinatiions combinations of 4 to 7 letters, chosen from a, b, c, d, e, f, g, with "either two or three letters between a and b".

 

[Alche]

Apparently Alche is NOT talking about "permutations" but about both permutations and combinatiions combinations of 4 to 7 letters, chosen from a, b, c, d, e, f, g, with "either two or three letters between a and b".

But in the question it is only mentioned about permutations of the letters, they have no where asked about the combination. I think I was not able to grasp what it was asking earlier, now I got it.

If am wrong then please correct me.