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 eq...

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

Product of any r consecutive natural numbers is always divisible by

For each n $$\in$$ N, 23n $$-$$ 1 is divisible by here N is a set of natural numbers.

If C0, C1, C2, ......, Cn denote the coefficients in the expansion of (1 + x)n then the value of C1 + 2C2 + 3C3 + ..... + nCn 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 5th and 6th 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_{2...

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

The number (101)100 $$-$$ 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...

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

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

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

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 ...

72n + 16n $$-$$1 (n$$ \in $$ N) is divisible by

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,

$$A = \left[ {\matrix{ 1 & 2 \cr 0 & 1 \cr } } \right]$$ then by the principle of mathematical induction, prove that $${A^n} = \left[ {\ma...

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 ...

Prove by induction that for all n $$\in$$ N, n2 + n is an even integer (n $$\ge$$ 1).

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