If the magnitude of the coefficient of x⁷ in the expansion of $${\left( {a{x^2} + {1 \over {bx}}} \right)^8}$$, where a, b are positive numbers, is equal to the magnitude of the coefficient of x⁷ in the expansion of $${\left( {ax + {1 \over {b{x^2}}}} \right)^8}$$, then a and b are connected by the relation

If $${}^{16}{C_r} = {}^{16}{C_{r + 1}}$$, then the value of $${}^r{P_{r - 3}}$$ is

The coefficient of x^$$-$$10 in $${\left( {{x^2} - {1 \over {{x^3}}}} \right)^{10}}$$ is

If C₀, C₁, C₂, ......, C_n denote the coefficients in the expansion of (1 + x)ⁿ then the value of C₁ + 2C₂ + 3C₃ + ..... + nC_n is

If the coefficients of x² and x³ in the expansion of (3 + ax)⁹ be same, then the value of a is

using binomial theorem, the value of (0.999)³ correct to 3 decimal places is

$$({2^{3n}} - 1)$$ will be divisible by $$(\forall n \in N)$$

If in the expansion (a $$-$$ 2b)ⁿ, the sum of the 5^th and 6^th term is zero, then the value of $${a \over b}$$ is

Sum of the last 30 coefficients in the expansion of (1 + x)⁵⁹, when expanded in ascending powers of x is

If $${(1 - x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + \,\,....\,\,{a_{2n}}{x^{2n}}$$, then the value of $${a_0} + {a_2} + {a_4} + \,\,....\,\,{a_{2n}}$$ is

The coefficient of xⁿ om the expansion of $${{{e^{7x}} + {e^x}} \over {{e^{3x}}}}$$ is

If A and B are coefficients of xⁿ in the expansions of (1 + x)²ⁿ and (1 + x)^{2n $$-$$ 1} respectively, then A/B is equal to

If n > 1 is an integer and x $$\ne$$ 0, then (1 + x)ⁿ $$-$$ nx $$-$$ 1 is divisible by

If $\left(1+x-2 x^2\right)^6=1+a_1 x+a_2 x^2+\ldots+a_{12} x^{12}$, then the value of $a_2+a_4+a_6+\ldots+a_{12}$ is

If $$\left(1+x+x^2+x^3\right)^5=\sum_\limits{k=0}^{15} a_k x^k$$ then $$\sum_\limits{k=0}^7(-1)^{\mathbf{k}} \cdot a_{2 k}$$ is equal to

The coefficient of $$a^{10} b^7 c^3$$ in the expansion of $$(b c+c a+a b)^{10}$$ is

The number of zeros at the end of $$\left| \!{\underline {\, {100} \,}} \right. $$ is

Show that, for a positive integer n, the coefficient of x^k (0 $$\le$$ k $$\le$$ n) in the expansion of 1 + (1 + x) + (1 + x)² + ....... + (1 + x)ⁿ is $${}^{n + 1}{C_{n - k}}$$.

If $$n$$ is a positive integer, the value of $$(2 n+1){ }^n C_0+(2 n-1){ }^n C_1+(2 n-3){ }^n C_2 +\ldots .+1 \cdot{ }^n C_n$$ is