Permutations an...

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is 

If $$S_n=\displaystyle\sum^{n}_{r=0}\dfrac{1}{^{n}C_r}$$ and $$t_n=\displaystyle\sum^n_{r=0}\dfrac{r}{^{n}C_r}$$, then $$\dfrac{t_n}{S_n}$$ is equal to?

$$2C_0 + \dfrac{2^2}{2} C_1 + \dfrac{2^3}{3} C_2 +........+ \dfrac{2^{11}}{11} C_{10}$$ is equal to

The number of 5 letter words formed  using letters of word "CALCULUS" is  

If $$^{2n}C_{3}:^{n}C_{2}::44:1$$, then the value of $$n$$ is

If m denotes the number of $$5$$ digit numbers if each successive digits are in their descending order of magnitude and n is the corresponding figure, when the digits are in their ascending order of magnitude then $$(m-n)$$ has the value?

$$10$$ IIT and $$2$$ NIT students sit at random in a row, and then number of ways in which exactly $$3$$ IIT students sit between $$2$$ NIT students is

$${ if}^{ n }{ c }_{ 10 }{ = }^{ n }{ c }_{ 14 }$$ then the value of n  is equal to

The sum $$^{ n }{ C }_{ 0 }+^{ n }{ C }_{ 1 }+^{ n }{ C }_{ 2 }+.......+^{ n }{ C }_{ n }$$ is equal to 

If $$\displaystyle\sum^m_{k=1}(k^2+1)k!=1999(2000!)$$, then m is?