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The number of ways can 7 boys be seated at a round table so that 2 particular boys are separated is :

$5!^{5} P_{2}$
$480$
$7! - 6! 2!$
6!2!

hint Hint:

Total number of ways in which they are separated = Total ways of arranging 7 boys at a round table - Total ways of arranging 2 boys sitting together

The correct answer is: $480$

Step by step solution :
We want to find the numbers of ways in which 7 boys can be seated on a round table when 2 particular boys are seperated.
Let us assume that two boys put together a single unit.
Now, we will arrange 6 entities in 6 places on a circular table.
When $n$ objects are arranged in $n$ places in a circular table, then there are $(n - 1)!$ ways.
That is, if there are 6 boys and 6 places, then there are $(6 - 1)! = 5!$ ways to arrange.
Now, the two boys that were fixed in a single place can also rearrange themselves.
The number of ways in which two boys can arrange among themselves is 2!
Next, to find the number of ways 7 boys can be seated at a round table so that 2 particular boys are next to each other is multiply 5! with 2!
That is, total ways possible are 5!2!.
Total ways of arranging 2 boys sitting together = 5! 2! = 240.
Total ways of arranging 7 boys at a round table = (7–1)! = 6! = 720

Total number of ways in which they are separated = Total ways of arranging 7 boys at a round table - Total ways of arranging 2 boys sitting together
Thus, Total number of ways in which they are separated = 720 - 240 = 480

When $n$ distinct objects are arranged in $n$ places, then the total number of ways possible are $n!$ But, when $n$ objects are to arranged in $n$ places on a circular table, there are $(n - 1)!$ ways.

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The number of ways can 7 boys be seated at a round table so that 2 particular boys are separated is :

6!2!

Total number of ways in which they are separated = Total ways of arranging 7 boys at a round table - Total ways of arranging 2 boys sitting together

The correct answer is:

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$5!^{5} P_{2}$
$480$
$7! - 6! 2!$
6!2!

The correct answer is: $480$

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The number of values of c such that st. line y = 4x + c touches the curve $fraction numerator x to the power of 2 end exponent over denominator 4 end fraction plus y to the power of 2 end exponent equals 1$ is

Tangents are drawn to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 5 end fraction equals 1$ at the ends of the latus rectum. The are of the quadrilateral so formed is

Tangents are drawn to the ellipse $fraction numerator x to the power of 2 end exponent over denominator 9 end fraction plus fraction numerator y to the power of 2 end exponent over denominator 5 end fraction equals 1$ at the ends of the latus rectum. The are of the quadrilateral so formed is

A tangent to the ellipse x² + 4y² = 4 meets the ellipse x² + 2y² = 6 at P and Q. The angle between the tangents at P and Q of the ellipse x² + 2y² = 6 is

A tangent to the ellipse x² + 4y² = 4 meets the ellipse x² + 2y² = 6 at P and Q. The angle between the tangents at P and Q of the ellipse x² + 2y² = 6 is