# Difficult Permutation-Combination Solved QuestionAptitude Discussion

 Q. There are 10 seats around a circular table. If 8 men and 2 women have to seated around a circular table, such that no two women have to be separated by at least one man. If $P$ and $Q$ denote the respective number of ways of seating these people around a table when seats are numbered and unnumbered, then $P : Q$ equals,
 ✖ A. $9 : 1$ ✖ B. $72 : 1$ ✔ C. $10 : 1$ ✖ D. $8 : 1$

Solution:
Option(C) is correct

Initially we look at the general case of the seats not numbered.

The total number of cases of arranging 8 men and 2 women, so that women are together:

⇒ $8×!2!$

The number of cases where in the women are not together:

$⇒ 9! - (8!×2!) = Q$

Now, when the seats are numbered, it can be considered to a linear arrangement and the number of ways of arranging the group such that no two women are together is:

⇒ $10! - (9!×2!)$

But the arrangements where in the women occupy the first and the tenth chairs are not favourable as when the chairs which are assumed to be arranged in a row are arranged in a circle, the two women would be sitting next to each other.

The number of ways the women can occupy the first and the tenth position:

$= 8!×2!$

The value of $P$,

$= 10! - (9!×2!) - (8!×2!)$

Thus $P : Q = \textbf{10 : 1}$

## (4) Comment(s)

Aamir Ansari
()

great solution! but at least correct the question.

Raj
()

Chaitana
()

I have a question.

Solution I get is 600:

0 1 2 3 4 5

first we select only from 5 digits because we cannot have zero as the first digit for five digit number.

first digit -> 5 ways

Repetition is not allowed

one of the digits from 1 2 3 4 5 is already selected. So, we have only 5 digits left because we can now use 0 as well

second digit - 5

third digit - 4

fourth digit - 3

fifth digit - 2

In total it would be: $5*5*4*3*2 = 600.$

What is wrong with this procedure?

Why do I get a different answer.

Subharthi
()

"such that no two women have to be separated by at least one man"

From this statement it seems that the question is asking for number of ways they can be seated such that two women are always together. but in the solution it has considered those cases when two women are not together.

To me this is not consistent with what has been asked. If the statement was in how many number of ways they can be seated such that two women are separated by at least one man then this solution was OK.

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