Placing symbols so that no row remains empty. 
Q.1.The symbols +, + , #, # , *, $ ($6$ in total) are to be placed in the squares of the given figure. Find the number of ways of placing the symbols so that no row (there are $5$ rows) remains empty.
I know combinatorics at undergrad level. This problems are from advanced challenging section.  

First, this problem is not as hard as it looks.  It appears reasonable to assume that the symbols are used up as they are placed.  For now, we can think of each symbol as being unique.  Since the symbols are placed "so that no row remains empty", and since there are 6 symbols, we have:
    6 ways to put a symbol into row 1
    5 ways to put a symbol into row 2
    ...
    2 ways to put a symbol into row 5

Note that only 3 of the rows has room for a final symbol.  Thus there are:
    3 ways to choose one of 3 rows to place the final symbol

We have over-counted due to the fact that there are duplicate #, # and +, +.  So we must divide by 2*2.
Finally the number of ways is given by:
(6*5*4*3*2 * 3) / (2*2) = 540

So the answer is 540 ways.
