Question

In how many ways can 3 girls and 3 boys be seated in a row of 6 chairs if 1 boy and 1 girl refused to sit next to each other?

Answer 1

First find the number of arrangements possible if the particular boy and the girl do not refuse to sit next to each other. Now, find the number of ways of arrange if this particular boy and the girl always sits together. For this, form their group and consider them and a single person and arrange 5 people at 5 places. Finally, subtract the number of arrangements in the latter case from the number of arrangements in the former case to get the answer.

Complete step-by-step solution:
Here there are 3 girls and 3 boys who are to be seated in 6 chairs with the condition that 1 boy and 1 girl refused to sit next to each other. According to the question it is already fixed that which boy and girl refused to six next to each other so we are not going to select them particularly.
Now, consider the case that this boy and girl do not refuse to sit together so we have a total of 6 people who need to be arranged at 6 places. So we get,
$\Rightarrow$ Number of arrangements possible = $6!$
Now, let us consider the case where we assume that this boy and the girl always sits together, so we can consider them as one unit or roughly we can consider them as a single person. Therefore, now we have 5 people that need to be arranged.
$\Rightarrow$ Number of arrangements possible = $5!$
Also, this particular boy and girls can be arranged among them in $2!$ ways. So we have an effective number of arrangements $=5!\times 2!$.
Therefore, the total number of arrangements where this boy and girl do not sit together $=6!-\left( 5!\times 2! \right)=480$.
Hence, there are 480 such arrangements possible.

Note: Note that if we will try to find the required arrangements directly then it will become a bit complicated as there are 6 places and to form arrangements according to the condition given is difficult. This is the reason we have followed the above process. Also note that we do not have to select the particular boy and the girl in $^{3}{{C}_{1}}{{\times }^{3}}{{C}_{1}}$ ways because they are already chosen and fixed in the question. If we will try to use the combination formula to select them then we will get the negative value in the answer which is not possible.