Question:
$300$ boys and $100$ girls were selected in JEE – 2007
from class – room programme. It was decided to take
a picture of the whole students in such a way that all
students to stand in a row, with boys standing in
decreasing order according to their height (assuming
all has distinct height) from left to right and girls
standing in increasing order according to their height
(assuming all has distinct height) from left to right.
In how may ways this can be done (The boys need
not stand together and girls need to stands together)?
My approach: Out of $400$ places in a row first we can choose $300$ places to arrange boys in descending order of the heights then in remaining hundred places we can arrange girls in descending order of the heights. I am getting $400 \choose 300$ am I correct.
Can you suggest alternate methods?
Best Answer
There is one way of arranging boys here: in decreasing order, from left to right. Since there are 300 boys, there are 301 places in which we can place the girls. Since the order of boys and girls is fixed, we cannot permute them. Therefore, there are only $\boxed{301}$ ways of doing this.