Question

Q11. The coefficient of x⁶ y³ in the expression (x + 2y)⁹ is?
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Answer 1

Coefficient of x⁶y³ is 672.

Step-by-step explanation:

General term of expansion (a+b)ⁿ is

$\bf \: T_{r+1} = \: \: ^nC_r \: \: \large \frak{ a ^{n−r} b ^r}$

For (x+2y)⁹,

Putting n =9, a=x, b=2y

$\bf \: T_{r+1} = \: \: ^{9} C_r (x) ^{9−r} (2y) ^r \\ \\ \bf \: T _{r+1} = \: \: ^{9} C_r (x) ^{9−r} .(y) ^r .(2) ^r$

Comparing with x⁶ y³ , we get, r = 3

Therefore,

$\bf \: T _{r+1} \\ \: \tt ^9C_3 (x)^9−3 .y³ .2³ \\ \tt\: \: 9! (2)³× x⁶ × y³) / (3!.6!) \\ \: \tt 672x⁶ y³$

Answer 2

Step-by-step explanation:

Coefficient of x⁶y³ is 672.

Step-by-step explanation:

General term of expansion (a+b)ⁿ is

\bf \: T_{r+1} = \: \: ^nC_r \: \: \large \frak{ a ^{n−r} b ^r}T

r+1

=

n

C

r

a

n−r

b

r

For (x+2y)⁹,

Putting n =9, a=x, b=2y

\begin{gathered} \bf \: T_{r+1} = \: \: ^{9} C_r (x) ^{9−r} (2y) ^r \\ \\ \bf \: T _{r+1} = \: \: ^{9} C_r (x) ^{9−r} .(y) ^r .(2) ^r\end{gathered}

T

r+1

=

9

C

r

(x)

9−r

(2y)

r

T

r+1

=

9

C

r

(x)

9−r

.(y)

r

.(2)

r

Comparing with x⁶ y³ , we get, r = 3

Therefore,

\begin{gathered} \bf \: T _{r+1} \\ \: \tt ^9C_3 (x)^9−3 .y³ .2³ \\ \tt\: \: 9! (2)³× x⁶ × y³) / (3!.6!) \\ \: \tt 672x⁶ y³\end{gathered}

T

r+1

9

C

3

(x)

9

−3.y³.2³

9!(2)³×x⁶×y³)/(3!.6!)

672x⁶y³