Question

5 boys and 3 girls are sitting in a row of 8 seats. Number of ways in which they can be seated so that not all the girls sit side by side, is?
A. 36000
B. 9080
C. 3960
D. 11600

Answer 1

Hint: We will first find the total number of ways in which all people can be arranged. Then, find the number of ways in which 3 girls can sit together by considering all three girls as a single unit. And at last, subtract the number of ways in which three girls together from the total number of ways in all persons can be arranged.

Complete step-by-step answer:
When there are $n$ objects and $n$ places, then the number of ways in which students can be arranged is $n!$.
There are total 8 person and 8 seats, then the total number of ways in which the person can be seated is $8!$
But, we need that not all the girls sit side by side.
We will first find the number of ways in which 3 girls can sit together.
Let us consider all three girls as a single unit.
Then there will be 6 people that need to arrange and the number of ways in arranging them is 6!.
And, the three girls can also arrange among themselves.
Then, the total number of ways three girls will sit together will be $3!\left( {6!} \right)$
But, we want to find the number of ways in which three girls cannot sit together.
We will subtract the number of ways in which three girls together from the total number of ways in all persons can be arranged.
$8! - 3!\left( {6!} \right)$
Also, we know \[n! = n.\left( {n - 1} \right).\left( {n - 2} \right).....3.2.1\].
Then, $8! = 8.7.6.5.4.3.2.1 = 40320$, $6! = 720$ and $3! = 6$
On substituting the values, we will get,
$40320 - \left( {720} \right)\left( 6 \right) = 40320 - 4320 = 36000$
Hence, there are 36,000 ways in which all people can be seated when all three girls do not sit together.
Thus, option A is correct.

Note: When there are $n$ distinct objects and needs to be arranged in $n$ places, then the number of ways in which these can be arranged is $n!$. Also, \[n! = n.\left( {n - 1} \right).\left( {n - 2} \right).....3.2.1\]. We only need cases when 3 girls are not sitting together, 2 girls still can sit together.