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Question: The coefficient of \[x^{7}\] in \[\left( 1-x-x^{2}+x^{3}\right)^{6} \] is A. -144 B. 144 C. -1...

The coefficient of x7x^{7} in (1−x−x2+x3)6\left( 1-x-x^{2}+x^{3}\right)^{6} is
A. -144
B. 144
C. -128
D. -142

Explanation

Solution

Hint: In this question it is given that we have to find the coefficient of x7x^{7} from the expression (1−x−x2+x3)6\left( 1-x-x^{2}+x^{3}\right)^{6} .
So to find the solution we need to know about binomial expansion, which is (1−a)n= nC0− nC1⋅a+ nC2⋅a2−…+(−1)n nCn⋅an\left( 1-a\right)^{n} =\ ^{n} C_{0}-\ ^{n} C_{1}\cdot a+\ ^{n} C_{2}\cdot a^{2}-\ldots +(-1)^{n}\ ^{n} C_{n}\cdot a^{n}.
So by this expression we are able to find the solution.

Complete step-by-step answer:
Here given expression,
(1−x−x2+x3)6\left( 1-x-x^{2}+x^{3}\right)^{6}
taking x2x^2 common from 3rd and 4th terms, we get.
=\left\\{ \left( 1-x\right) -x^{2}\left( 1-x\right) \right\\}^{6}
=\left\\{ \left( 1-x\right) \left( 1-x^{2}\right) \right\\}^{6}
=(1−x)6(1−x2)6=\left( 1-x\right)^{6} \left( 1-x^{2}\right)^{6}
By expanding the 1st and 2nd terms by the use of Binomial expansion, we get,
=(6C0− 6C1x+ 6C2x2−…+ 6C6x6)(6C0− 6C1(x2)+ 6C2(x2)2− 6C3(x2)3+…+ 6C6(x2)6)=\left( {}^{6}C_{0}-\ ^{6} C_{1}x+\ ^{6} C_{2}x^{2}-\ldots +\ ^{6} C_{6}x^{6}\right) \left( {}^{6}C_{0}-\ ^{6} C_{1}\left( {}x^{2}\right) +\ ^{6} C_{2}\left( x^{2}\right)^{2} -\ ^{6} C_{3}\left( x^{2}\right)^{3} +\ldots +\ ^{6} C_{6}\left( x^{2}\right)^{6} \right)
On simplifying the power of ‘x’, we get,
=(6C0− 6C1x+ 6C2x2−…+ 6C6x6)(6C0− 6C1x2+ 6C2x4− 6C3x6+…+ 6C6x12)=\left( {}^{6}C_{0}-\ ^{6} C_{1}x+\ ^{6} C_{2}x^{2}-\ldots +\ ^{6} C_{6}x^{6}\right) \left( {}^{6}C_{0}-\ ^{6} C_{1}x^{2}+\ ^{6} C_{2}x^{4}-\ ^{6} C_{3}x^{6}+\ldots +\ ^{6} C_{6}x^{12}\right)
Now we have to find the coefficient of x7x^{7},
Therefore, the coefficient of x7x^{7} is
=(coefficient of x×coefficient of x6)+(coefficient of x3×coefficient of x4)+(coefficient of x5×coefficient of x2)=\left( \text{coefficient of} \ x\times \text{coefficient of} \ x^{6}\right) +\left( \text{coefficient of} \ x^{3}\times \text{coefficient of} \ x^{4}\right) +\left( \text{coefficient of} \ x^{5}\times \text{coefficient of} \ x^{2}\right)
=\left\\{ \left( -{}^{6}C_{1}\right) \times \left( -{}^{6}C_{3}\right) \right\\} +\left\\{ \left( -{}^{6}C_{3}\right) \times \left( {}^{6}C_{2}\right) \right\\} +\left\\{ \left( -{}^{6}C_{5}\right) \times \left(-{}^{6}C_{1}\right) \right\\}
=(6C1 6C3)−(6C3 6C2)+(6C5 6C1)=\left( {}^{6}C_{1}\ ^{6} C_{3}\right) -\left( {}^{6}C_{3}\ ^{6} C_{2}\right) +\left( {}^{6}C_{5}\ ^{6} C_{1}\right)
We know that nCr{}^{n}C_{r} can be written as, nCr=n!r!⋅(n−r)!{}^{n}C_{r}=\dfrac{n!}{r!\cdot \left( n-r\right) !}
6!1!(6−1)!×6!3!(6−3)!−6!3!(6−3)!×6!2!(6−2)!+6!5!(6−5)!×6!1!(6−1)!\dfrac{6!}{1!\left( 6-1\right) !} \times \dfrac{6!}{3!\left( 6-3\right) !} -\dfrac{6!}{3!\left( 6-3\right) !} \times \dfrac{6!}{2!\left( 6-2\right) !} +\dfrac{6!}{5!\left( 6-5\right) !} \times \dfrac{6!}{1!\left( 6-1\right) !}
=6!1!⋅5!×6!3!⋅3!−6!3!⋅3!×6!2!⋅4!+6!5!⋅1!×6!1!⋅5!=\dfrac{6!}{1!\cdot 5!} \times \dfrac{6!}{3!\cdot 3!} -\dfrac{6!}{3!\cdot 3!} \times \dfrac{6!}{2!\cdot 4!} +\dfrac{6!}{5!\cdot 1!} \times \dfrac{6!}{1!\cdot 5!}
=6⋅5!5!×6⋅5⋅4⋅3!3⋅2⋅1⋅3!−6⋅5⋅4⋅3!3⋅2⋅1⋅3!×6⋅5⋅4!2⋅1⋅4!+6⋅5!5!×6⋅5!5!=\dfrac{6\cdot 5!}{5!} \times \dfrac{6\cdot 5\cdot 4\cdot 3!}{3\cdot 2\cdot 1\cdot 3!} -\dfrac{6\cdot 5\cdot 4\cdot 3!}{3\cdot 2\cdot 1\cdot 3!} \times \dfrac{6\cdot 5\cdot 4!}{2\cdot 1\cdot 4!} +\dfrac{6\cdot 5!}{5!} \times \dfrac{6\cdot 5!}{5!}
=6×5×4−5×4×6⋅52+6×6=6\times 5\times 4-5\times 4\times \dfrac{6\cdot 5}{2} +6\times 6
=(6×5×4)−(5×4×3×5)+(6×6)=(6\times 5\times 4)-(5\times 4\times 3\times 5)+(6\times 6)
=120−300+36=120-300+36
=−144=-144
Therefore the coefficient of x7x^{7} is -144.
Hence the correct option is option A.

Note: While solving this type of question you need to know that nCr{}^{n}C_{r} can be written as, nCr=n!r!⋅(n−r)!{}^{n}C_{r}=\dfrac{n!}{r!\cdot \left( n-r\right) !}
Where n!=n⋅(n−1)⋅(n−2)⋯3⋅2⋅1n!=n\cdot \left( n-1\right) \cdot \left( n-2\right) \cdots 3\cdot 2\cdot 1 and
n!=n⋅(n−1)!n!=n\cdot \left( n-1\right) !
Also while finding the coefficient of 7 ^{7} you no need to multiply each and every term, it will make the solution lengthy.