The coefficient of $x^{5} i n {(1 - x - x^{2} + x^{3})}^{6}$ is

-36

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

-216

No worries! We‘ve got your back. Try BYJU‘S free classes today!

204

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B -36
$[(1 - x) + x^{2} (x - 1)]^{6}$
$\Rightarrow (1 - x)^{6} (I) (1 - x)^{6} (I I)$
$\Rightarrow^{6} C_{0} 1^{6} (- x)^{o} +^{6} C_{1}$
For $x^{5}$ $\Rightarrow$ combinations possible:-
$0^{t h}$ term of (I) & $5^{t h}$ term of (II)
$\Rightarrow^{6} C_{0} 1^{6} (- x^{2})^{0} \times^{6} C_{5} 1 \cdot (- x)^{5} = - 6 x^{5}$
$1^{s t}$ term of $(1)$ & $3^{r d}$ term of (II)
$\Rightarrow^{6} C_{1} 1^{5} \cdot (- x^{2})^{1} \times^{6} C_{3} \cdot 1^{3} \cdot (- x)^{3}$
$= \frac{- 6 \times 6 \times 5 \times 4}{6} x^{5} = - 120$ x
$2^{n d}$ term of (I) & $1^{s t}$ term of (II)
$\Rightarrow^{6} C_{2} 1^{4} \cdot (- x^{2})^{2} \times^{6} C_{1} \cdot 1^{5} \cdot (- x)^{1}$
$= \frac{- 6 \times 5}{2} \times 6 x^{5} = - 30 x^{5}$
$∴$ Coefficient of $x^{5} = - 6 - 120 - 90 = - 216$ .

Suggest Corrections

Similar questions

The coefficient of $x^{7}$ in $(1 - x - x^{2} + x^{3})^{6}$ is

x^{2}

{(x^{2} + \frac{1}{x^{3}})}^{6}

x

(1 - x - x^{2} + x^{3})^{6}

x^{7}

{(1 - x + x^{2} + x^{3})}^{6}

x^{7}

(1 - x - x^{2} + x^{3})^{6}

Join BYJU'S Learning Program