Find the term independent of \(x\), \(x \neq 0\), in the expansion of \(\left(\dfrac{3x^{2}}{2}-\dfrac{1}{3x}\right)^{15}\).
\(\dfrac{1001}{2592}\)
If the term free from \(x\) in the expansion of \(\left(\sqrt{x}-\dfrac{k}{x^{2}}\right)^{10}\) is \(405\), find the value of \(k\).
\(k=\pm 3\)
Find the coefficient of \(x\) in the expansion of \((1-3x+7x^{2})(1-x)^{16}\).
\(-19\)
Find the term independent of \(x\) in the expansion of \(\left(3x-\dfrac{2}{x^{2}}\right)^{15}\).
\(-3003\cdot 3^{10}\cdot 2^{5}\) (i.e. \(-3003\,(3^{10})(2^{5})\))
Find the middle term(s) in the following expansions:
(i) \(\left(\dfrac{x}{a}-\dfrac{a}{x}\right)^{10}\)
(ii) \(\left(3x-\dfrac{x^{3}}{6}\right)^{9}\)
(i) middle term is the term with index \(k=5\): \(-252\).
(ii) the two middle terms are:
for \(k=4\): \(\dfrac{189}{8}x^{17}\),
for \(k=5\): \(-\dfrac{21}{16}x^{19}\).
Find the coefficient of \(x^{15}\) in the expansion of \((x-x^{2})^{10}\).
\(-252\)
Find the coefficient of \(\dfrac{1}{x^{17}}\) in the expansion of \(\left(x^{4}-\dfrac{1}{x^{3}}\right)^{15}\).
\(-1365\)
Find the sixth term of the expansion \(\left(\dfrac{1}{y^{2}}+\dfrac{1}{x^{3}}\right)^{n}\), if the binomial coefficient of the third term from the end is \(45\).
From \(\binom{n}{2}=45\) we get \(n=10\). Sixth term (\(k=5\)) is \(\displaystyle \binom{10}{5}\,\dfrac{1}{y^{10}}\dfrac{1}{x^{15}}=252\,x^{-15}y^{-10}\).
Find the value of \(r\), if the coefficients of the \((2r+4)\)-th and \((r-2)\)-th terms in the expansion of \((1+x)^{18}\) are equal.
\(r=6\)
If the coefficients of the second, third and fourth terms in the expansion of \((1+x)^{2n}\) are in A.P., show that \(2n^{2}-9n+7=0\).
Let the coefficients be \(C_{1}=\binom{2n}{1}=2n\), \(C_{2}=\binom{2n}{2}=n(2n-1)\), \(C_{3}=\binom{2n}{3}=\dfrac{(2n)(2n-1)(2n-2)}{6}\). A.P. condition: \(2C_{2}=C_{1}+C_{3}\). Substituting and simplifying gives \(2n^{2}-9n+7=0\).
Find the coefficient of \(x^{4}\) in the expansion of \((1+x+x^{2}+x^{3})^{11}\).
Number of weak compositions of 4 into 11 parts (each \(\le 3\)) equals \(\binom{4+11-1}{11-1}=\binom{14}{10}=\binom{14}{4}=1001\).
If \(p\) is a real number and if the middle term in the expansion of \(\left(\dfrac{p}{2}+2\right)^8\) is 1120, find \(p\).
\(p=\pm 2\)
Show that the middle term in the expansion of \(\left(x-\dfrac{1}{x}\right)^{2n}\) is \(\dfrac{1\times3\times5\times\dots\times(2n-1)}{n!}\times(-2)^n\).
Find \(n\) in the binomial \(\left(\sqrt[3]{2}+\dfrac{1}{\sqrt[3]{3}}\right)^n\) if the ratio of the 7th term from the beginning to the 7th term from the end is \(\dfrac{1}{6}\).
\(n=9\)
In the expansion of \((x+a)^n\) if the sum of odd terms is denoted by \(O\) and the sum of even terms by \(E\), prove that:
(i) \(O^2-E^2=(x^2-a^2)^n\)
(ii) \(4OE=(x+a)^{2n}-(x-a)^{2n}\)
If \(x^p\) occurs in the expansion of \(\left(x^2+\dfrac{1}{x}\right)^{2n}\), prove that its coefficient is \(\dbinom{2n}{\dfrac{2n+p}{3}}\) (for integral \(\dfrac{2n+p}{3}\)).
Find the term independent of \(x\) in the expansion of \(\left(\dfrac{3}{2}x^2-\dfrac{1}{3x}\right)^9\).
The total number of terms in the expansion of \( (x + a)^{100} + (x - a)^{100} \) after simplification is
50
202
51
none of these
Given the integers \(r > 1, n > 2\), and coefficients of \((3r)^{th}\) and \((r+2)^{nd}\) terms in the binomial expansion of \((1 + x)^{2n}\) are equal, then
\(n = 2r\)
\(n = 3r\)
\(n = 2r + 1\)
none of these
The two successive terms in the expansion of \((1 + x)^{24}\) whose coefficients are in the ratio 1:4 are
3rd and 4th
4th and 5th
5th and 6th
6th and 7th
The coefficient of \(x^{n}\) in the expansion of \((1 + x)^{2n}\) and \((1 + x)^{2n-1}\) are in the ratio
1 : 2
1 : 3
3 : 1
2 : 1
If the coefficients of 2nd, 3rd and the 4th terms in the expansion of \((1 + x)^{n}\) are in A.P., then value of \(n\) is
2
7
11
14
Choose the correct statement regarding the binomial identities (question text from source).
Option A
Option B
Option C
Option D
Choose the correct statement regarding the binomial identities (question text from source).
Option A
Option B
Option C
Option D
The largest coefficient in the expansion of \((1 + x)^{30}\) is ________.
\(^{30}C_{15}\)
The number of terms in the expansion of \((x + y + z)^n\) is ________.
\(\dfrac{(n+1)(n+2)}{2}\)
In the expansion of \(\left(x^2 - \dfrac{1}{x^2}\right)^{16}\), the value of constant term is ________.
\(^{16}C_{8}\)
If the seventh terms from the beginning and the end in the expansion of \(\left(\sqrt[3]{2} + \dfrac{1}{\sqrt[3]{3}}\right)^n\) are equal, then \(n\) equals ________.
12
The coefficient of \(a^{-6} b^{4}\) in the expansion of \(\left(\dfrac{1}{a} - \dfrac{2b}{3}\right)^{10}\) is ________.
\(\dfrac{1120}{27} a^{-6} b^{4}\)
Middle term in the expansion of \((a^3 + ba)^{28}\) is ________.
\(^{28}C_{14} a^{56} b^{14}\)
The ratio of the coefficients of \(x^p\) and \(x^q\) in the expansion of \((1 + x)^{p+q}\) is ________.
1
The position of the term independent of \(x\) in the expansion of \(\left(\dfrac{x}{3} + \dfrac{3}{2x^2}\right)^{10}\) is ________.
Third term
If \(25^{15}\) is divided by 13, the remainder is ________.
12
The sum of the series \(\sum_{r=0}^{10} \binom{20}{r}\) is \(2^{19} + \dfrac{\binom{20}{10}}{2}\).
False
The expression \(7^9 + 9^7\) is divisible by 64.
True
The number of terms in the expansion of \([(2x + y^3)^4]^7\) is 8.
False
The sum of coefficients of the two middle terms in the expansion of \((1 + x)^{2n - 1}\) is equal to \(\binom{2n - 1}{n}\).
False
The last two digits of the number \(3^{400}\) are 01.
True
If the expansion of \((x - \dfrac{1}{x^2})^{2n}\) contains a term independent of \(x\), then \(n\) is a multiple of 2.
False
The number of terms in the expansion of \((a + b)^n\) where \(n \in \mathbb{N}\) is one less than the power \(n\).
False