If $${{}^{20}{C_r}}$$ is the co-efficient of xr in the expansion of (1 + x)20, then the value of $$\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}} $$ is equal to :
If the coefficients of x7 in $${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$$ and x$$-$$7 in $${\left( {{x} - {1 \over {bx^2}}} \right)^{11}}$$, b $$\ne$$ 0, are equal, then the value of b is equal to :
A
2
B
$$-$$1
C
1
D
$$-$$2
Explanation
Coefficient of x7 in $${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$$ :
General Term = $${}^{11}{C_r}{({x^2})^{11 - r}}.{\left( {{1 \over {bx}}} \right)^r}$$
= $${}^{11}{C_r}{x^{22 - 3r}}.{1 \over {{b^r}}}$$
$$22 - 3r = 7$$
$$r = 5$$
$$\therefore$$ Required Term = $${}^{11}{C_5}.{1 \over {{b^5}}}.{x^7}$$
Coefficient of x$$-$$7 in $${\left( {x - {1 \over {b{x^2}}}} \right)^{11}}$$ :
General Term = $${}^{11}{C_r}{(x)^{11 - r}}.{\left( { - {1 \over {b{x^2}}}} \right)^r}$$