Question:
The coefficient of $x^7$ in the expansion of $(1- x - x^2 + x^3 )^6$ is
The coefficient of $x^7$ in the expansion of $(1- x - x^2 + x^3 )^6$ is
Updated On: Jul 5, 2022
- -132
- -144
- 132
- 144
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
$(1 - x - x^2 + x^3)^6 = [(1- x) - x^2 (1 - x)]^6$
$ = (1- x)^6 (1 - x^2)^6$
$= (1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6)$
$ \times (1 - 6x^2 + 15x^4 - 20x^6 + 15x^8 - 6x^{10} + x^{12})$
Coefficient of $x^7 = (- 6) (- 20) + (- 20)(15) + (- 6) (-6) = - 144$
Was this answer helpful?
0
0
Top Questions on Binomial theorem
- The sum of the coefficients of \( x^{2/3} \) and \( x^{-2/5} \) in the binomial expansion of \[ \left( x^{2/3} + \frac{1}{2} x^{-2/5} \right)^9 \] is:
- JEE Main - 2024
- Mathematics
- Binomial theorem
- Let $m$ and $n$ be the coefficients of the seventh and thirteenth terms respectively in the expansion of \[\left( \frac{1}{3}x^{\frac{1}{3}} + \frac{1}{2x^{\frac{2}{3}}} \right)^{18}.\]Then \[\left(\frac{n}{m}\right)^{\frac{1}{3}}\]is:
- JEE Main - 2024
- Mathematics
- Binomial theorem
- If the term independent of \(x\) in the expansion of \[ \left( \sqrt{ax^2} + \frac{1}{2x^3} \right)^{10} \] is 105, then \(a^2\) is equal to:
- JEE Main - 2024
- Mathematics
- Binomial theorem
- Let $\alpha = \sum_{r=0}^n (4r^2 + 2r + 1) \binom{n}{r}$ and $\beta = \left( \sum_{r=0}^n \frac{\binom{n}{r}}{r+1} \right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$, then the value of $n$ is _____.
- JEE Main - 2024
- Mathematics
- Binomial theorem
- Let $0 \leq r \leq n$. If \[^nC_{r+1} : ^nC_r : ^nC_{r-1} = 55 : 35 : 21,\]then $2n + 5r$ is equal to:
- JEE Main - 2024
- Mathematics
- Binomial theorem
View More Questions
Questions Asked in AIEEE exam
- This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. It is not possible to make a sphere of capacity $1$ farad using a conducting material. It is possible for earth as its radius is $6.4 \times 10^6\,m$.
- AIEEE - 2012
- electrostatic potential and capacitance
- A steel wire can sustain $100\, kg$ weight without breaking. If the wire is cut into two equal parts, each part can sustain a weight of
- AIEEE - 2012
- mechanical properties of solids
- The limiting line in Balmer series will have a frequency of (Rydberg constant, $R_{\infty}=3.29\times10^{15}$ cycles/s)
- AIEEE - 2012
- Electromagnetic Spectrum
- Which of the plots shown in the figure represents speed (v) of the electron in a hydrogen atom as a function of the principal quantum number (n)?
- The ratio of number of oxygen atoms (O) in 16.0 g ozone $(O_3), \,28.0\, g$ carbon monoxide $(CO)$ and $16.0$ oxygen $(O_2)$ is (Atomic mass: $C = 12,0 = 16$ and Avogadro?? constant $N_A = 6.0 x 10^{23}\, mol^{-1}$)
- AIEEE - 2012
- Mole concept and Molar Masses
View More Questions
Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.