# If n boys and n girls sit along a line alternately in x ways and along a circle alternately in y ways such that x = 12y then n is equal to:

$

{\text{a}}{\text{. 6}} \\

{\text{b}}{\text{. 8}} \\

{\text{c}}{\text{. 9}} \\

{\text{d}}{\text{. 12}} \\

$

Last updated date: 30th Mar 2023

•

Total views: 310.2k

•

Views today: 2.86k

Answer

Verified

310.2k+ views

Hint: - Number of ways to sit along a circle by $n$ persons is$\left( {n - 1} \right)!$

Number of ways to sit along a line by n boys$ = n! $.

And the number of ways to sit along a line by n girls$ = n! $.

$\therefore $Starting from boy the number of ways to sit along a line by $n$ boys and $n$ girls alternately is $n! \times n! $.

Now, starting from girls, the number of ways to sit along a line by $n$ boys and $n$ girls alternately is $n! \times n! $.

Therefore total number of ways to sit along a line by $n$ boys and $n$ girls alternately

$

\left( x \right) = n! \times n! + n! \times n! \\

\Rightarrow x = 2 \times n! \times n! \\

$

Now, in circle starting does not matter because in the circle there are no starting and end points.

Therefore total no of ways to sit along a circle by $n$ boys and $n$ girls alternately

$ \Rightarrow y = \left( {n - 1} \right)! \times n! $

Now according to question it is given that $x = 12y$

$

\Rightarrow 2 \times n! \times n! = 12 \times \left( {n - 1} \right)! \times n! \\

\Rightarrow n! = 6 \times \left( {n - 1} \right)! \\

$

As we know that $n! = n\left( {n - 1} \right)!$

$

\Rightarrow n\left( {n - 1} \right)! = 6 \times \left( {n - 1} \right)! \\

\Rightarrow n = 6 \\

$

Hence, $n = 6$is the required answer.

$\therefore $Option (a) is correct.

Note: -In such types of questions first find out the total number of ways to sit along a line by $n$ boys and $n$ girls alternately and total number of ways to sit along a circle by $n$ boys and $n$ girls alternately, then equate them according to given condition then, we will get the required answer.

Number of ways to sit along a line by n boys$ = n! $.

And the number of ways to sit along a line by n girls$ = n! $.

$\therefore $Starting from boy the number of ways to sit along a line by $n$ boys and $n$ girls alternately is $n! \times n! $.

Now, starting from girls, the number of ways to sit along a line by $n$ boys and $n$ girls alternately is $n! \times n! $.

Therefore total number of ways to sit along a line by $n$ boys and $n$ girls alternately

$

\left( x \right) = n! \times n! + n! \times n! \\

\Rightarrow x = 2 \times n! \times n! \\

$

Now, in circle starting does not matter because in the circle there are no starting and end points.

Therefore total no of ways to sit along a circle by $n$ boys and $n$ girls alternately

$ \Rightarrow y = \left( {n - 1} \right)! \times n! $

Now according to question it is given that $x = 12y$

$

\Rightarrow 2 \times n! \times n! = 12 \times \left( {n - 1} \right)! \times n! \\

\Rightarrow n! = 6 \times \left( {n - 1} \right)! \\

$

As we know that $n! = n\left( {n - 1} \right)!$

$

\Rightarrow n\left( {n - 1} \right)! = 6 \times \left( {n - 1} \right)! \\

\Rightarrow n = 6 \\

$

Hence, $n = 6$is the required answer.

$\therefore $Option (a) is correct.

Note: -In such types of questions first find out the total number of ways to sit along a line by $n$ boys and $n$ girls alternately and total number of ways to sit along a circle by $n$ boys and $n$ girls alternately, then equate them according to given condition then, we will get the required answer.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?