The coefficient of x 7 in ( 1 - x + 2 x 3 ) 10 is _________ . [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
The coefficient of $x^{7}$ in ${(1 - x + 2 x^{3})}^{10}$ is _________ . [2023]

Ans.
(960)

General term is given by

$\frac{10!}{r_{1}! r_{2}! r_{3}!} {(1)}^{r_{1}} {(- x)}^{r_{2}} {(2 x^{3})}^{r_{3}}$ i.e., $\frac{10!}{r_{1}! \cdot r_{2}! \cdot r_{3}!} {(- 1)}^{r_{2}} {\cdot (2)}^{r_{3}} x^{r_{2} + 3 r_{3}}$

where $r_{1} + r_{2} + r_{3} = 10$ and $r_{2} + 3 r_{3} = 7$

Possible values of $(r_{1}, r_{2}, r_{3})$ are $(3, 7, 0), (5, 4, 1)$ and $(7, 1, 2)$

Required coefficient

$= \frac{10!}{3! 7!} {(- 1)}^{7} + \frac{10!}{5! 4!} {(- 1)}^{4} (2) + \frac{10!}{7! 2!} {(- 1)}^{1} {(2)}^{2}$

$= - 120 + 2520 - 1440 = 960$

Want Us to Email you About Special Offers & Updates?