Question

Among all the possible permutations that can be made by all the characters from the series of words "COMPUTER", how many different ways that always and only three characters are in between 'P' and 'R''? Show both your work and the exact number.

Answer 1

In the word 'COMPUTER' there are 8 letters. Total number of permutations of all the letters= 8! =40320.

Exactly 3 characters are there in between 'P' and 'R'. Please look at the table. First row is indicating the position of letters.

In 2nd row we can see 'P' is sitting 1st position, 'R' in 5th position.

In 3rd row we can see 'P' is in 2nd position , 'R' in 6th position.

In 4th row we can see 'P' is in 3rd position , 'R' in 7th position.

In 5th row we can see 'P' is in 4th position , 'R' in 8th position.

So tatal 4 cases can be there when 'P' is coming 3 position before than 'R'.

1	2	3	4	5	6	7	8
P				R
	P				R
		P				R
			P				R

Similarly 4 cases can be there when 'R' is coming 3 position before than 'P' like below-

1	2	3	4	5	6	7	8
R				P
	R				P
		R				P
			R				P

So , there are total 8 cases when there are exactly 3 letters between 'P' and 'R'.

now there are 8-2=6 remaining positions and 6 letters. For each of 8 cases for 'P' & 'R' remaining 6 letters can be permutted 6! ways i.e. 720 ways.

So, Total number of cases= 8*720 =5760 where only 3 characters are in between 'P' and 'R'.

Answer is 5760.

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