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.  

Position would be $(1,1,1,1,2)$ or $(1,1,2,1,1)$ or $(2,1,1,1,1)$ for five rows respectively.
Positions to which can beselected in $\binom 32$ for the row with $2$ elements and $3\times3$ for other two.
Now treating each symbol as a unique entity, arrangement can be done in $6!$ ways. Now to cancel  multiples we divide by $2!2!$ one each for + and #. So ways are:
$$3\times\left[\binom 32\times3\times3\right]\times \frac14\times6!=14580\text{ ways}$$
