Permutations an...

$$14X's$$ have to be placed in the squares of the above figure such that each row contains at least one $$X$$. In how many ways can this be done?

If all the permutations of the letters in the word 'OBJECT' are arranged (and numbered serially) in alphabetical order as in a dictionary, then the $$717^{th}$$ word is

The value of $$^{47}C_4 + \displaystyle\sum_{r=1}^{5}{^{52-r}C_3}$$=

If the difference of the number of arrangements of three things from a certain number of dissimilar things and the number of selections of the same number of things from them exceeds $$ 100$$, then the least number of dissimilar things is

$$p$$ is a prime number and $$n < p < 2n$$. If $$N=^{2n}C_n$$, then 

The rank of the word $$NUMBER$$ obtained, if the letters of the word $$NUMBER$$ are written in all possible orders and these words are written out as in a dictionary is

$$^nC_r + \displaystyle\sum_{j=0}^{3}{^{n+j}C_{r+1+j}} = $$_________________

The number of factors (excluding $$1$$ and the expression itself) of the product of $$a^7b^4c^3def$$ where $$a,b,c,d,e,f$$ are all prime numbers is 

If $$\space ^{n+1}C_{r+1}:  ^nC_r :  ^{n-1}C_{r-1} = 11:6:3$$, then find the values of $$n$$ and $$r$$.

There are three papers of $$100$$ marks each in an examination. In how many ways can a student get $$150$$ marks such that he gets at least $$60%$$ in two papers?