Question

Five boys and three girls are sitting in a row of eight seats. In how many ways can they be seated so that not all the girls sit side by side?

Answer 1

Hint: For this we calculate the solution of given problem in two steps, in first step we calculate total number of arrangement of all eight persons without any restriction and in step two we calculate arrangement of all three girls together and then finding difference of the two results obtained, we will get total number of ways in which all girls can’t sit together and hence required solution of the problem.

Complete step-by-step answer:
Total number of boys = $ 5 $
Number of girls = $ 3 $
Therefore, total number of person required to be seated side by side = $ 8 $
To find a solution to this problem we find the solution in two steps.
In step one, we calculate the total number of ways in which eight people can arrange them on either seat without any restriction. This can be done in:
\[
  8!\,\,ways \\
  or \\
  8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\
   = 40320 \;
 \]
So, the correct answer is “ 40320 ”.

In the second step we calculate the number of arrangements in which all three girls sit together. This can be done in:
 $
  ^3{P_3}{ \times ^5}{P_5}{ \times ^6}{P_1} \\
   = 3!\, \times \,5!\, \times \,6 \\
   = 6 \times 120 \times 6 \\
   = 36 \times 120 \\
   = 4320 \;
  $
Now, to find the total number of ways in which all three girls not sit together can be find by calculating the difference of total number of ways in which all eight people sit with any restriction and total numbers of ways in which all three girls sit together. This can be done in:
 $
  40320 - 4320 \\
   = 36000 \;
  $
Hence, from above we see that total numbers of ways in which five boys and three girls are sitting in a row, such that all three girls are not sit together = $ 36000 $
So, the correct answer is “ $ 36000 $ ”.

Note: In this type of problem some time students use the block method. In this method student make arrangement of eight blocks in which they make arrangement of girls at either at even places or at odd places and then make arrangement of boys and hence calculate total number of arrangements but this is a wrong way as in this no girl sit together but in given problem all three girls not to sit together but two can. This is why students will get different answers in the block method and hence wrong answers to the problem.