Permutations an...

2. $$^n$$C$$_0$$ + 2$$^2$$ . $$\dfrac{^nC_1}{2}$$ + 2$$^3$$ . $$\dfrac{^nC_2}{3}$$ +.......+ 2$${n + 1}$$ . $$\dfrac{^nC_n}{n + 1}$$ = 

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

The sum $$\sum _{ r=1 }^{ n }{ r.{  }^{ 2n }{ C }_{ r } } $$ is equal to.

The number of ways in which three girls and ten boys can be seated in two vans, each having numbered seats, three in the front and four at the back is

The total no. of six digit numbers $${ x }_{ 1 }{ x }_{ 2 }{ x }_{ 3 }{ x }_{ 4 }{ x }_{ 5 }{ x }_{ 6 }$$ having the property $${ x }_{ 1 }<{ x }_{ 2 }\le { x }_{ 3 }<{ x }_{ 4 }<{ x }_{ 5 }\le { x }_{ 6 }$$, is equal to

If $$^nC_r$$ denotes the number of combinations of n things taken r things at a time, then the expression $$^nC_{r+1} + ^nC_{r-1}+2 ^nC_r is$$

If $$^{ n }{ C }_{ 3 }=^{ n }{ C }_{ 2 }$$, then n is equal to

$$If\,a\, = {\,^m}{C_2},\,\,then{\,^a}{C_{2\,}}\,is\,equal\,to$$

$$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +\dfrac { 2{ C }_{ 2 } }{ { C }_{ 1 } } +...\dfrac { 3{ C }_{ 3 } }{ { C }_{ n-1 } } =$$

If $$\sum\limits_{r = 0}^n {\left( {\frac{{r + 2}}{{r + 1}}} \right)} \,{\,^n}{C_r} = \frac{{{2^8} - 1}}{6},$$ then 'n' is