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## 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

^{n}P

_{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.