JEE Mains · Maths · STD 11 - 7. binomial theoram
The coefficient of \(x^{256}\) in the expansion of \((1-x)^{101}\left(x^{2}+x+1\right)^{100}\) is:
- A \({-}^{100} \mathrm{C}_{16}\)
- B \(^{100} \mathrm{C}_{16}\)
- C \(^{100} \mathrm{C}_{15}\)
- D \(-{ }^{100} \mathrm{C}_{15}\)
Answer & Solution
Correct Answer
(C) \(^{100} \mathrm{C}_{15}\)
Step-by-step Solution
Detailed explanation
\(y=(1-x)(1-x)^{100}\left(x^{2}+x+1\right)^{100}\) \(y=(1-x)\left(x^{3}-1\right)^{100}\) \(y=\left(x^{3}-1\right)^{100}-x\left(x^{3}-1\right)^{100}\) Coff. Of \(x^{256}\) in \(y=-\) coff of \(x^{255}\) in \(\left(x^{3}-1\right)^{100}\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(f : (-1, 1) \to R\) be a function defined by \(f\left( x \right) = \left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}\). if \(K\) be the set of all points at which \(f\) is not differentiable, then \(K\) has exactlyJEE Mains 2019 Hard
- In a bolt factory, machines \(A, B\) and \(C\) manufacture respectively \(20 \%, 30 \%\) and \(50 \%\) of the total bolts. Of their output \(3,4\) and \(2\) percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine \(C\) isJEE Mains 2023 Hard
- The value of the integral \(\int_{0}^{\pi}|\sin 2 x| dx\) isJEE Mains 2021 Easy
- The area of the region (in sq. units), in the first quadrant bounded by the parabola \(y = 9x^2\) and the lines \(x = 0,y = 1\) and \(y = 4,\) isJEE Mains 2013 Hard
- The value of \(\lambda\) and \(\mu\) such that the system of equations \(x+y+z=6,3 x+5 y+5 z=26, x+2 y+\lambda z=\mu\) has no solution, are :JEE Mains 2021 Medium
- Let \(\sqrt 3 \hat i + j,\hat i + \sqrt 3 \hat j\) and \(\beta \hat i + \left( {1 + \beta } \right)\hat j\) respectively be the position vectors of the points \(A,B\) and \(C\) with respect to the origin \(O\). If the distance of \(C\) from the bisector of the acute angle between \(OA\) and \(OB\) is \(\frac{3}{{\sqrt 2 }}\) , then the sum of all possible values of \(\beta \) isJEE Mains 2019 Hard
More PYQs from JEE Mains
- Let \(z\) be a complex number such that \(\left| z \right| + z = 3 + i\) (where \(i = \sqrt { - 1} \)). Then \(\left| z \right|\) is equal toJEE Mains 2019 Hard
- Let there be three independent events \(E _{1}, E _{2}\) and \(E _{3}\). The probability that only \(E _{1}\) occurs is \(\alpha\), only \(E _{2}\) occurs is \(\beta\) and only \(E _{3}\) occurs is \(\gamma .\) Let \('p'\) denote the probability of none of events occurs that satisfies the equations \((\alpha-2 \beta) p =\alpha \beta\) and \((\beta-3 \gamma) p =2 \beta \gamma .\) All the given probabilities are assumed to lie in the interval \((0,1)\) Then, \(\frac{\text { Probability of occurrence of } E _{1}}{\text { Probability of occurrence of } E _{3}}\) is equal to ..........JEE Mains 2021 Hard
- Let \(S\) be the set of all \(a \in N\) such that the area of the triangle formed by the tangent at the point \(P ( b , c ), b , c \in N\), on the parabola \(y ^2=2 ax\) and the lines \(x=b, y=0\) is \(16\) unit \(^2\), then \(\sum_{\text {aes }} a\) is equal to \(..........\).JEE Mains 2023 Hard
- Let \(\dfrac{x^2}{f(a^2+7a+3)} + \dfrac{y^2}{f(3a+15)} = 1\) represent an ellipse with major axis along \(y\)-axis, where \(f\) is a strictly decreasing positive function on \(\mathbb{R}\). If the set of all possible values of \(a\) is \(\mathbb{R} - [\alpha, \beta]\), then \(\alpha^2+\beta^2\) is equal to:JEE Mains 2026 Hard
- If \(\mathrm{b}\) is very small as compared to the value of \(\mathrm{a}\), so that the cube and other higher powers of \(\frac{b}{a}\) can be neglected in the identity \(\frac{1}{a-b}+\frac{1}{a-2 b}+\frac{1}{a-3 b} \ldots .+\frac{1}{a-n b}=\alpha n+\beta n^{2}+\gamma n^{3}\), then the value of \(\gamma\) is:JEE Mains 2021 Hard
- Let the line \(x+y=1\) meet the circle \(x^2+y^2=4\) at the points A and B . If the line perpendicular to \(A B\) and passing through the mid point of the chord \(A B\) intersects the circle at \(C\) and \(D\), then the area of the quadrilateral ADBC is equal to :JEE Mains 2025 Medium