Question

In how many ways can 6 boys and 5 girls be arranged for groups to photograph if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?

Answer 1

Hint: Place all five girls in one row and another 6 boys in the other row. Now, use formula n! to arrange n different things. And hence calculate the arrangements of boys and girls individually and hence calculate the total arrangements with the help of the arrangements of boys and girls separately.

Complete step-by-step answer:

Let the 6 boys be represented by ${{B}_{1}},{{B}_{2}},{{B}_{3}},{{B}_{4}},{{B}_{5}},{{B}_{6}}$ and 5 girls ${{G}_{1}},{{G}_{2}},{{G}_{3}},{{G}_{4}},{{G}_{5}}$ . Now, it is given that girls are sitting on the chairs in a row and boys are going to stand behind them. It means arrangement will look,
On chairs$\to {{G}_{1}},{{G}_{2}},{{G}_{3}},{{G}_{4}},{{G}_{5}}\leftarrow$ first row
Stand $\to {{B}_{1}},{{B}_{2}},{{B}_{3}},{{B}_{4}},{{B}_{5}},{{B}_{6}}\leftarrow$ second row
So, we can observe that we do not need to arrange all 11 girls and boys as we are not arranging them in a single row, we have two rows and have to arrange boys and girls separately. So, we know that the arrangement of n different things can be given by n! = 1.2.3…………n.
So, we can arrange 6 boys by 6! boys in the second row standing behind the girls.
So, total arrangement for boys = 6! = 720
Now, similarly we can arrange 5 girls in the first row by 5! Ways. So, we get
Total arrangement for girls = 5! = 120
Now, we know that total arrangements of boys and girls can be given by multiplying the arrangements of girls and boys. So, the total way to arrange for the photograph would be given by $720\times 120=86400$ .
Hence, there are 86400 ways to get a photograph such that girls will sit on chairs in one row and boys will stand behind the girls in another row.
Hence, the answer is 86400.

Note: One measure mistake may happen here that students will do arrangement of all 11 girls and boys and may give an answer as 11!. But it is wrong we are not arranging them in a single row, we are arranging them in two separate rows. So, we do need to calculate permutations separately. So, be clear with this part.
One may go wrong while calculating the arrangement for boys and girls. He or she may add the arrangements of girls and boys calculated separately, which is wrong as both the permutations cannot be added to get arrangements of both of them combined. So, be clear with the fundamental concepts of multiplication and addition in permutations and combinations.