Question:
The number of ways in which $5$ ladies and $7$ gentlemen can be seated in a round table so that no two ladies sit together, is
The number of ways in which $5$ ladies and $7$ gentlemen can be seated in a round table so that no two ladies sit together, is
Updated On: Jun 22, 2024
- $ \frac{7}{2}{{(720)}^{2}} $
- $ 7{{(360)}^{2}} $
- $ 7{{(720)}^{2}} $
- $ 720 $
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The Correct Option is A
Solution and Explanation
First we fix the alternate position of 7 gentlemen in a round table by 6! ways. There are seven positions between the gentlemen in which 5 ladies can be seated in $ ^{7}{{P}_{5}} $ ways.
$ \therefore $ Required number of ways
$=6!\times \frac{7!}{2!} $
$=\frac{7}{2}{{(720)}^{2}} $
$ \therefore $ Required number of ways
$=6!\times \frac{7!}{2!} $
$=\frac{7}{2}{{(720)}^{2}} $
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Concepts Used:
Permutations
A permutation is an arrangement of multiple objects in a particular order taken a few or all at a time. The formula for permutation is as follows:
\(^nP_r = \frac{n!}{(n-r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
Types of Permutation
- Permutation of n different things where repeating is not allowed
- Permutation of n different things where repeating is allowed
- Permutation of similar kinds or duplicate objects