Binomial Theorem

The sum of the series
2.20C0
+ 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to :

The sum of the co-efficients of all even
degree terms in x in the expansion of
\({\left( {x + \sqrt {{x^3} - 1} } \right)^6}\) + \({\left( {x - \sqrt {{x^3} - 1} } \right)^6}\), (x > 1) is equal to:

If the fourth term in the binomial expansion of
\({\left( {\sqrt {{x^{\left( {{1 \over {1 + {{\log }_{10}}x}}} \right)}}} + {x^{{1 \over {12}}}}} \right)^6}\) is equal to 200, and x > 1,
then the value of x is :

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of \({\left( {{2^{1/3}} + {1 \over {2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}}\) is :

The total number of irrational terms in the binomial expansion of (71/5 – 31/10)60 is :

The coefficient of x18 in the product
(1 + x) (1 – x)10 (1 + x + x2)9
is :

The term independent of x in the expansion of
\(\left( {{1 \over {60}} - {{{x^8}} \over {81}}} \right).{\left( {2{x^2} - {3 \over {{x^2}}}} \right)^6}\) is equal to :

If 20C1 + (22) 20C2 + (32) 20C3 + ..... + (202
)
20C20 = A(2\(\beta\)), then the ordered pair (A, \(\beta\)) is equal to :

The value of r for which 20Cr 20C0 + 20Cr\(-\)1 20C1 + 20Cr\(-\)2 20C2 + . . . . .+ 20C0 20Cr  is maximum, is

The sum of the real values of x for which the middle term in the binomial expansion of \({\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}\) equals 5670 is :

Let Sn = 1 + q + q2 + . . . . . + qn and Tn = 1 + \(\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}\) + . . . . . .+ \({\left( {{{q + 1} \over 2}} \right)^n}\) where q is a real number and q \(\ne\) 1. If...

Let (x + 10)50 + (x \(-\) 10)50 = a0 + a1x + a2x2 + . . . . + a50x50, for all x \(\in\) R; then \({{{a_2}} \over {{a_0}}}\) is equal to

If  \({\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}\)  then k is equal to

If the third term in the binomial expansion of \({\left( {1 + {x^{{{\log }_2}x}}} \right)^5}\) equals 2560, then a possible value of x is -

The positive value of \(\lambda\) for which the co-efficient of x2
in the expression x2 \({\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}\) is 720, is -

If the coefficients of x2
and x3
are both zero, in the expansion of the expression (1 + ax + bx2
) (1 – 3x)15 in
powers of x, then the ordered pair (a,b) is equal to :

The smallest natural number n, such that the coefficient of x in the expansion of \({\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}\) is nC23, is :

The coefficient of x2 in the expansion of the product
(2\(-\)x2) .((1 + 2x + 3x2)6 + (1 \(-\) 4x2)6) is :

If n is the degree of the polynomial,
\({\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8} +\) \({\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}\)
and m is the coefficient of xn in it, ...

The coefficien of x10 in the expansion of (1 + x)2(1 + x2)3(1 + x3)4 is equal to :

The sum of the co-efficients of all odd degree terms in the expansion of
\({\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5}\), \(\left( {x > 1} \right)\) is

The coefficient of x−5 in the binomial expansion of
\({\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + 1}} - {{x - 1} \over {x - {x^{{1 \over 2}}}}}} \right)^{10}},\) where x \(\ne\) 0, 1, is :

If (27)999 is divided by 7, then the remainder is :

The value of \(\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} - {}^{10}{C_3}} \right)\)
\(\left( {{}^{21}{C_4} - {}^{10}{C_4}} \right)\)$$ + .... + \left( {{}^{21}{C_{10}} ...

For x \(\in\) R, x \(\ne\) -1,
if (1 + x)2016 + x(1 + x)2015 + x2(1 + x)2014 + . . . . + x2016 =
\(\sum\limits_{i = 0}^{2016} {{a_i}} \,{x^i},\,\,\) then a17 is equal to :

If the coefficients of x−2 and x−4 in the expansion of \({\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \right)^{18}},\left( {x > 0} \right),\) are m and n respectively, then \({m \over n}\) is equal to :

If the number of terms in the expansion of \({\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n},\,x \ne 0,\) is 28, then the sum of the coefficients of all the terms in this expansion, is :

The sum of coefficients of integral power of \(x\) in the binomial expansion \({\left( {1 - 2\sqrt x } \right)^{50}}\) is :

If the coefficints of \({x^3}\) and \({x^4}\) in the expansion of \(\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}\) in powers of \(x\) are both zero, then \(\left( {a,\,b} \right)\) is equal to:

The term independent of \(x\) in expansion of
\({\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}\) is

If \(n\) is a positive integer, then \({\left( {\sqrt 3 + 1} \right)^{2n}} - {\left( {\sqrt 3 - 1} \right)^{2n}}\) is :

The coefficient of \({x^7}\) in the expansion of \({\left( {1 - x - {x^2} + {x^3}} \right)^6}\) is

Let \({s_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right){}^{10}} {C_j}\),
\({{s_2} = \sum\limits_{j = 1}^{10} {} } j.{}^{10}{C_j}\) and \({{s_3} = \sum\limits_{j = 1}^{10} {{j^2}.{}^{10}{C_j}.} }\)
Statement-1 : $${{S_3} = 55 \times...

The remainder left out when \({8^{2n}} - {\left( {62} \right)^{2n + 1}}\) is divided by 9 is :

Statement - 1 : \(\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.}\)
Statement - 2 : $$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r} = {{\left( {1 + x} \right)}^n} + nx{{\left( ...

The sum of the series \({}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .....\, - \,.....\, + {}^{20}{C_{10}}\) is

In the binomial expansion of \({\left( {a - b} \right)^n},\,\,\,n \ge 5,\) the sum of \({5^{th}}\) and \({6^{th}}\) terms is zero, then \(a/b\) equals

If the expansion in powers of \(x\) of the function \({1 \over {\left( {1 - ax} \right)\left( {1 - bx} \right)}}\) is \({a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}.....\) then \({a_n}\) is

For natural numbers \(m\) , \(n\), if \({\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\, = 1 + {a_1}y + {a_2}{y^2} + ..........\) and \({a_1} = {a_2} = 10,\) then \(\left( {m,\,n} \right)\) is

If the coefficient of \({x^7}\) in \({\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}\) equals the coefficient of \({x^{ - 7}}\) in \({\left[ {ax - \left( {{1 \over {b{x^2}}}} \right)} \right]^{11}}\), then \(a\) and \(b\)...

If the coefficients of rth, (r+1)th, and (r + 2)th terms in the binomial expansion of \({{\rm{(1 + y )}}^m}\) are in A.P., then m and r satisfy the equation

If \(x\) is so small that \({x^3}\) and higher powers of \(x\) may be neglected, then \({{{{\left( {1 + x} \right)}^{{3 \over 2}}} - {{\left( {1 + {1 \over 2}x} \right)}^3}} \over {{{\left( {1 - x} \right)}^{{1 \over 2}}}}}\) may be approxi...

The value of \(\,{}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}} {C_3}\) is

The coefficient of the middle term in the binomial expansion in powers of \(x\) of \({\left( {1 + \alpha x} \right)^4}\) and \({\left( {1 - \alpha x} \right)^6}\) is the same if \(\alpha\) equals

If \({S_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}}} \,\,and\,\,{t_n} = \sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}},\,}\)then \({{{t_{ n}}} \over {{S_n}}}\) is equal to

The coefficient of \({x^n}\) in expansion of \(\left( {1 + x} \right){\left( {1 - x} \right)^n}\) is

The number of integral terms in the expansion of \({\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}\) is

If \(x\) is positive, the first negative term in the expansion of \({\left( {1 + x} \right)^{27/5}}\) is

The coefficients of \({x^p}\) and \({x^q}\) in the expansion of \({\left( {1 + x} \right)^{p + q}}\) are

If the sum of the coefficients in the expansion of \(\,{\left( {a + b} \right)^n}\) is 4096, then the greatest coefficient in the expansion is