Permutations an...

If the letters of the word "VARUN" are written in all possible ways and then are arranged as in a dictionary, then the rank of the word VARUN is?

Garlands are formed using 6 red roses and 6 yellow roses of different sizes. The number of arrangements in garland which have red roses and yellow roses come alternately is

The value of  $$\sum\limits_{r = 0}^{10} {\left( {\matrix{   {10}  \cr    r  \cr } } \right)} \left( {\matrix{   {15}  \cr    {14 - r}  \cr } } \right)$$ is equal to

If $$f(x)=1-x+x^2-x^3+....-x^{15}+x^{16}+x^{17}$$, then the coefficient of $$x^2$$ in $$f(x-1)$$ is?

The product of five consecutive numbers is always divisible by ?

A person tries to form as many different parties as he can, out of his $$20$$ friends. Each party should consist of the same number. How many friends should be invited at a time? In how many of these parties would the same friends be found?

Let $$(1+x)^n=C_0+C_1x+C_2x^2+....+C_nX^n$$.(where $$C_r=$$ $$^{n}C_r$$). On the basis of information, answer the following question.
$$2(C_2)-4(C_4)+6(C_6)$$_______ is?

Let $$5 < n_1 < n_2 < n_3 < n_4$$ be integers such that $$n_1+n_2+n_3+n_4=35$$. The number of such distinct arrangements $$(n_1, n_2, n_3, n_4)$$.

The value of $${ _{  }^{ 13 }{ C } }_{ 2 }+{ _{  }^{ 13 }{ C } }_{ 3 }+{ _{  }^{ 13 }{ C } }_{ 4 }+....+{ _{  }^{ 13 }{ C } }_{ 13 }$$ is 

In an examination there are three multiple choice questions and each question has $$4$$ choices. Number of ways in which a student can fail to get all answer correct is?