Permutations an...

The value of the expression $${  }^{ n+1 }{ C }_{ 2 }+2[{  }^{ 2 }{ C }_{ 2 }+{  }^{ 3 }{ C }_{ 2 }+{  }^{ 4 }{ C }_{ 2 }+.....+{  }^{ n }{ C }_{ 2 }]$$ is

The value of $${  }^{ 15 }{ C }_{ 8 }+{  }^{ 15 }{ C }_{ 9 }-{  }^{ 15 }{ C }_{ 6 }-{  }^{ 15 }{ C }_{ 7 }$$ is :

Number of positive integers  $$n ,$$  less than  $$17 ,$$  for which  $$n ! + ( n + 1 ) ! + ( n + 2 ) !$$  is an integral multiple of  $$49$$  is

$$\left( \begin{matrix} 51 \\ 3 \end{matrix} \right) +\left( \begin{matrix} 50 \\ 3 \end{matrix} \right) +\left( \begin{matrix} 49 \\ 3 \end{matrix} \right) +\left( \begin{matrix} 48 \\ 3 \end{matrix} \right) +\left( \begin{matrix} 47 \\ 3 \end{matrix} \right) +\left( \begin{matrix} 47 \\ 4 \end{matrix} \right) $$ is equal to (where $$\left( \begin{matrix} n \\ r \end{matrix} \right) $$ denotes $$_{  }^{ n }{ { C }_{ r } }$$

The value of $$\frac{{^{11}{C_0}}}{1} + \frac{{^{11}{C_1}}}{2} + \frac{{^{11}{C_2}}}{3} + ....... + \frac{{^{11}{C_{11}}}}{{12}}$$ will be 

How many combinations can be formed of 8 counters marked 1 2 ..... 8 taking 4 at a time there being at lest one odd and even numbered in each combination? 

$$\sum _{ r=1 }^{ n }{ \left\{ \sum _{ { r }_{ 1 }=0 }^{ r-1 }{ ^{ n }{ C }_{ r }^{ n }{ C }_{ { r }_{ 1 } }{ 2 }^{ { r }_{ 1 } } }  \right\}  } =$$

Which of the following is not true?

$$
^{47} C_{r}+\sum_{j=1}^{5}(52-1) C_{3}=
 $$

If $$\sum\limits_{r = m}^n {^r{C_m}{ = ^{n + 1}}{C_{m + 1,}}} $$ then $$\sum\limits_{r = m}^n {{{\left( {n - r + 1} \right)}^r}{C_m}} $$ is equal to