Binomial Theorem

The sum, of the coefficients of the first 50 terms in the binomial expansion of \((1-x)^{100}\), is equal to

If \(\frac{1}{n+1}{ }^{n} \mathrm{C}_{n}+\frac{1}{n}{ }^{n} \mathrm{C}_{n-1}+\ldots+\frac{1}{2}{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{0}=\frac{1023}{10}\) then \(n\) is equal to :

The number of integral terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}\) is equal to ___________.

The mean of the coefficients of \(x, x^{2}, \ldots, x^{7}\) in the binomial expansion of \((2+x)^{9}\) is ___________.

If the \(1011^{\text {th }}\) term from the end in the binominal expansion of \(\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{2022}\) is 1024 times \(1011^{\text {th }}\)R term from the beginning, then \(|x|\) is equal to

The sum of the coefficients of three consecutive terms in the binomial expansion of \((1+\mathrm{x})^{\mathrm{n}+2}\), which are in the ratio \(1: 3: 5\), is equal to :

The coefficient of \(x^7\) in \({(1 - x + 2{x^3})^{10}}\) is ___________.

If the coefficient of \({x^7}\) in \({\left( {ax - {1 \over {b{x^2}}}} \right)^{13}}\) and the coefficient of \({x^{ - 5}}\) in \({\left( {ax + {1 \over {b{x^2}}}} \right)^{13}}\) are equal, then \({a^4}{b^4}\) is equal to :

If the coefficients of \(x\) and \(x^{2}\) in \((1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}\) are 4 and \(-\)5 respectively, then \(2 p+3 q\) is equal to :

Let the number \((22)^{2022}+(2022)^{22}\) leave the remainder \(\alpha\) when divided by 3 and \(\beta\) when divided by 7. Then \(\left(\alpha^{2}+\beta^{2}\right)\) is equal to :

For two positive real numbers a and b such that \({1 \over {{a^2}}} + {1 \over {{b^3}}} = 4\), then minimum value of the constant term in the expansion of \({\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}}\) is :

If the constant term in the expansion of
\({\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}\) is 2k.l, where l is an odd integer, then the value of k is equal to:

Let the coefficients of x\(-\)1 and x\(-\)3 in the expansion of \({\left( {2{x^{{1 \over 5}}} - {1 \over {{x^{{1 \over 5}}}}}} \right)^{15}},x > 0\), be m and n respectively. If r is a positive integer such that $$m{n^2} = {}^{15}{C_r}\,.\,...

Let n \(\ge\) 5 be an integer. If 9n \(-\) 8n \(-\) 1 = 64\(\alpha\) and 6n \(-\) 5n \(-\) 1 = 25\(\beta\), then \(\alpha\) \(-\) \(\beta\) is equal to

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}\), in the increasing powers of \(\frac{1}{\sqrt[4]{3}}\) be $$\sqrt...

\(\text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }\) ________.

\(\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)\) is equal to

The number of positive integers k such that the constant term in the binomial expansion of \({\left( {2{x^3} + {3 \over {{x^k}}}} \right)^{12}}\), x \(\ne\) 0 is 28 . l, where l is an odd integer, is ______________.

If \(\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} }\), where ...

The term independent of x in the expansion of \((1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0\) is :

The remainder when \(7^{2022}+3^{2022}\) is divided by 5 is :

If \(1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})\) is equal to \(2^{\mathrm{n}} \cdot \mathrm{m}\), where \(\mathrm{m}\) is odd, then $$\mathrm{n...

Let the coefficients of the middle terms in the expansion of \(\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}\) and \(\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0\), respectively form the first three terms of an A.P. If d is...

If the coefficient of x10 in the binomial expansion of \({\left( {{{\sqrt x } \over {{5^{{1 \over 4}}}}} + {{\sqrt 5 } \over {{x^{{1 \over 3}}}}}} \right)^{60}}\) is \({5^k}\,.\,l\), where l, k \(\in\) N and l is co-prime to 5, then k is eq...

If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of \({\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}\) is 939, then the sum of all the possible integral values of n is _________.

The remainder when \((2021)^{2022}+(2022)^{2021}\) is divided by 7 is

Let for the \(9^{\text {th }}\) term in the binomial expansion of \((3+6 x)^{\mathrm{n}}\), in the increasing powers of \(6 x\), to be the greatest for \(x=\frac{3}{2}\), the least value of \(\mathrm{n}\) is \(\mathrm{n}_{0}\). If $$\mathrm...

The remainder when (2021)2023 is divided by 7 is :

If \(\left( {{}^{40}{C_0}} \right) + \left( {{}^{41}{C_1}} \right) + \left( {{}^{42}{C_2}} \right) + \,\,.....\,\, + \,\,\left( {{}^{60}{C_{20}}} \right) = {m \over n}{}^{60}{C_{20}}\) m and n are coprime, then m + n is equal to ___________...

If the coefficients of \(x\) and \(x^{2}\) in the expansion of \((1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}, \mathrm{p}, \mathrm{q} \leq 15\), are \(-3\) and \(-5\) respectively, then the coefficient of \(x^{3}\) is equal to _____________.

\(\sum\limits_{\matrix{ {i,j = 0} \cr {i \ne j} \cr } }^n {{}^n{C_i}\,{}^n{C_j}}\) is equal to

Let Cr denote the binomial coefficient of xr in the expansion of \({(1 + x)^{10}}\). If for \(\alpha\), \(\beta\) \(\in\) R, \({C_1} + 3.2{C_2} + 5.3{C_3} +\) ....... upto 10 terms $$ = {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\le...

If \({1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}\), then the remainder when K is divided by 6 is :

If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of \({\left( {2{x^3} + {3 \over x}} \right)^{10}}\) is \({5^{10}} - \beta \,.\,{3^9}\), then \(\beta\) is equal to ____________.

The coefficient of x101 in the expression \({(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}\), x > 0, is

If the maximum value of the term independent of \(t\) in the expansion of \(\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{\mathrm{t}}\right)^{15}, x \geqslant 0\), is \(\mathrm{K}\), then \(8 \mathrm{~K}\) is equal to ___...

The remainder when \((11)^{1011}+(1011)^{11}\) is divided by 9 is

The remainder when 32022 is divided by 5 is :

The remainder on dividing 1 + 3 + 32 + 33 + ..... + 32021 by 50 is _________.

If \(\left( {{{{3^6}} \over {{4^4}}}} \right)k\) is the term, independent of x, in the binomial expansion of \({\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}\), then k is equal to ___________.

If the coefficient of a7b8 in the expansion of (a + 2b + 4ab)10 is K.216, then K 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 possible value of 'x', for which the ninth term in the expansion of \({\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}} \right\}^{10}}\) in the increasing powers of...

\(\sum\limits_{k = 0}^{20} {{{\left( {{}^{20}{C_k}} \right)}^2}}\) is equal to :

3 \(\times\) 722 + 2 \(\times\) 1022 \(-\) 44 when divided by 18 leaves the remainder __________.

Let m, n\(\in\)N and gcd (2, n) = 1. If $$30\left( {\matrix{
{30} \cr
0 \cr

} } \right) + 29\left( {\matrix{
{30} \cr
1 \cr

} } \right) + ...... + 2\left( {\matrix{
{30} \cr
{28} \cr

} } \right) + 1\left(...

The maximum value of the term independent of 't' in the expansion of \({\left( {t{x^{{1 \over 5}}} + {{{{(1 - x)}^{{1 \over {10}}}}} \over t}} \right)^{10}}\) where x\(\in\)(0, 1) is :

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 :

Let \(\left( {\matrix{ n \cr k \cr } } \right)\) denotes \({}^n{C_k}\) and $$\left[ {\matrix{
n \cr
k \cr

} } \right] = \left\{ {\matrix{
{\left( {\matrix{
n \cr
k \cr

} } \right),} & {if\,0 \le k \le ...

The term independent of 'x' in the expansion of \({\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}\), where x \(\ne\) 0, 1 is equal to ______________.