Permutations an...

$${ if }^{ n }{ c }_{ r }{ + }^{ n }{ c }_{ r+1 }={ n+1 }_{ { C }_{ x } }$$ then the value of x is equal to

The value of $$\sum _{ r=0 }^{ n }{ \sum _{ s=0 }^{ n }{ (r+s){ C }_{ r }{ C }_{ s } }  } ,$$ is 

If $$n\ \in\ N$$ & $$n$$ is even, then $$\dfrac {1}{1\ .\ (n-1)\ !}+\dfrac {1}{3\ !(n-3)\ !}+\dfrac {1}{5\ !\  (n-5)\ !}+....+\dfrac {1}{(n-1)\ !\ 1\ !}=$$ 

$${ if } ^ { n }{ C }_{ 3 }{ + }^{ n }{ C }_{ 4 }>{ n+1 }_{ C_{ 3 } }$$ then

if$$^{ 2017 }{ c }_{ 0 }{ + }^{ 2017 }{ c }_{ 1 }{ + }^{ 2017 }{ c }_{ 2 }+.....{ + }^{ 2017 }{ c }_{ 1008 }={ \lambda  }^{ 2 }(\lambda >0),$$ then remainder when $$\lambda$$ is divided by 33 is

Value of $$\sum _{ r=1 }^{ n }{ \left( \sum _{ m=0 }^{ r }{ {  }_{  } }  \right)  } ^nC_r,^rC_m)$$ is equal to 

If $$(1+x)^n=C_0+C_1x+C_2x^2+....+C_nX^n$$, then the value of $$C^2_0+\dfrac{C^2_1}{2}+\dfrac{C^2_2}{3}+....+\dfrac{C^2_n}{n+1}$$ is equal to?

If $$(1+x)^{n}=C_{0}+C_{1}x+C_{2}x^{2}+...+C_{n}x^{n}$$ then the value of $$1^{2}C_{1}+2^{2}C_{2}+3^{2}C_{3}+...+n^{2}C_{n}$$ is

The value of $${^{50}C_4}+\sum^6_{r=1}{^{56-r}C_3}$$ is?

If n is odd natural number, then $$\sum _{ r=0 }^{ n }{ \cfrac { { (-1) }^{ r } }{ ^{ n }{ C }_{ r } }  } $$ equals