Permutations an...

If $$(1+x)^{15}=C_{0}+C_{1}xC_{2}x^{2}+...+C_{15}x^{15},$$ then $$^{15}C_{0}^{2}- ^{15}C_{1}^{2}+^{15}C_{2}^{2}- ^{15}C_{2}^{3}+... ^{15}C_{15}^{2}$$ is equal to

How many different $$5$$ letter sequences can be made using the letters $$A,B,C,D$$ with repetition such that the sequence does not include the word $$BAD$$?

If $$^{n}C_{1}=^{n}C_{2}$$, then $$n$$ is equal to 

$${ C }_{ 0 }+({ C }_{ 0 }+{ C }_{ 1 })+({ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 })+......+({ C }_{ 0 }+{ C }_{ 1 }+........+{ C }_{ n })=$$

If $$^{n}C_{r}=\dfrac {n\ !}{r\ !(n-r)\ !}$$ then the sum of the series $$1+^{n}C_{1}+^{n+1}C_{2}+^{n+2}C_{3}+.....+^{n+r-1}C_{r}$$ is equal to

If the second term of the expansion $$\left[a^{1/13}+\dfrac{a}{\sqrt{a^{-1}}}\right]^n$$ is $$14a^{5/2}$$ then the value of $$\dfrac{^{n}C_3}{^{n}C_2}$$ is

If $$\dfrac{c_0}{2}-\dfrac{c_1}{3}+\dfrac{c_2}{4}-.....+(-1)^n\dfrac{c_n}{n+2}=\dfrac{1}{2013\times2014}$$ then n =

$$\sum _{ r=0 }^{ n }{ r\left( n-r \right) \left( _{  }^{ n }{ { C }_{ r } } \right) ^{ 2 } } $$ is equal to 

The value $$\sum _{ r=0 }^{ 10 }{ r_{  }^{ 10 }{ C_{ r } } } .{ \left( -2 \right)  }^{ 10-r }$$ is

If $$^{50}C_{r}=C_{r}$$ then the value of $$C_{0}^{2}+\frac{1}{2}C_{1}^{2}+\frac{1}{3}C_{2}^{2}+\frac{1}{4}C_{3}^{2+....+\frac{1}{50}^{101}C_{51}}$$