Binomial Theorem

The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)^{15}$ is equal to

If $\sum_{r=0}^{2 n} a_r(x-2)^r=\sum_{r=0}^{2 n} b_r(x-3)^r$ and $a_k=1 \forall k \geq 1$, then the value of $\frac{b_n}{{ }^{2 n+1} C_{n+1}}$ is

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 \(n\) 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

The coefficient of \(a^{10} b^7 c^3\) in the expansion of \((b c+c a+a b)^{10}\) 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 number of zeros at the end of \(\left| \!{\underline {\, {100} \,}} \right.\) is

The remainder when \({7^{{7^{{7^{{{..}^7}}}}}}}\) (22 time 7) is divided by 48 is

The coefficient of a3b4c5 in the expansion of (bc + ca + ab)6 is

For x\(\in\)R, x \(\ne\) \(-\)1, if \({(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ..... + {x^{2016}} = \sum\limits_{i = 0}^{2016} {{a_i}\,.\,{x^i}}\), then a17 is equal to

If c0, c1, c2, ......, c15 are the binomial coefficients in the expansion of (1 + x)15, then the value of \({{{c_1}} \over {{c_0}}} + 2{{{c_2}} \over {{c_1}}} + 3{{{c_3}} \over {{c_2}}} + ... + 15{{{c_{15}}} \over {{c_{14}}}}\) is

The number of irrational terms in the expansion of \({\left( {{3^{{1 \over 8}}} + {5^{{1 \over 4}}}} \right)^{84}}\) is

The number (101)100 \(-\) 1 is divisible by

If n is even positive integer, then the condition that the greatest term in the expansion of (1 + x)n may also have the greatest coefficient, is

Let \({(1 + x + {x^2})^9} = {a_0} + {a_1}x + {a_2}{x^2} + ... + {a_{18}}{x^{18}}\). Then,

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

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

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

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

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

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

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

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

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

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

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

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

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...