In expansion of $(x+2y)^{9}$, coefficient of $x^{6}y^{3}$ is

Q: In expansion of $(x+2y)^{9}$, coefficient of $x^{6}y^{3}$ is

To find the coefficient of $x^6y^3$ in the expansion of $(x + 2y)^9$, we use the binomial theorem. The binomial theorem states that: \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \] In this case, we have $(x + 2y)^9$, so $n = 9$, $x$ is $x$, and $y$ is $2y$. We are looking for the term where the power of $x$ is 6 and the power of $y$ is 3. According to the binomial theorem, the general term in the expansion is given by: \[ \binom{9}{k} x^{9-k} (2y)^k \] We need the term where $x^{9-k} = x^6$, which implies $9-k = 6$. Solving for $k$, we get: \[ k = 9 - 6 = 3 \] Now, substitute $k = 3$ into the general term: \[ \binom{9}{3} x^{9-3} (2y)^3 = \binom{9}{3} x^6 (2^3 y^3) \] Calculate $\binom{9}{3}$: \[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \] Calculate $2^3$: \[ 2^3 = 8 \] Now, substitute these values back into the term: \[ 84 \times x^6 \times 8 \times y^3 = 84 \times 8 \times x^6 y^3 \] Calculate the coefficient: \[ 84 \times 8 = 672 \] Thus, the coefficient of $x^6 y^3$ is 672. Therefore, the correct answer is Option 1: 672. **Explanation for Incorrect Options:** - **Option 2 (572):** This option is incorrect because the calculation of the coefficient using the binomial theorem and the powers of 2 results in 672, not 572. - **Option 3 (472):** This option is incorrect for the same reason as above. The correct calculation yields 672, not 472. - **Option 4 (272):** This option is incorrect because it significantly underestimates the coefficient. The correct calculation, as shown, results in 672. In conclusion, the correct coefficient of $x^6 y^3$ in the expansion of $(x + 2y)^9$ is indeed 672, which supports Option 1 as the correct answer.

Question

Accepted Answer

To find the coefficient of $x^6y^3$ in the expansion of $(x + 2y)^9$, we use the binomial theorem. The binomial theorem states that:

$$
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
$$

In this case, we have $(x + 2y)^9$, so $n = 9$, $x$ is $x$, and $y$ is $2y$. We are looking for the term where the power of $x$ is 6 and the power of $y$ is 3.

According to the binomial theorem, the general term in the expansion is given by:

$$
\binom{9}{k} x^{9-k} (2y)^k
$$

We need the term where $x^{9-k} = x^6$, which implies $9-k = 6$. Solving for $k$, we get:

$$
k = 9 - 6 = 3
$$

Now, substitute $k = 3$ into the general term:

$$
\binom{9}{3} x^{9-3} (2y)^3 = \binom{9}{3} x^6 (2^3 y^3)
$$

Calculate $\binom{9}{3}$:

$$
\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84
$$

Calculate $2^3$:

$$
2^3 = 8
$$

Now, substitute these values back into the term:

$$
84 \times x^6 \times 8 \times y^3 = 84 \times 8 \times x^6 y^3
$$

Calculate the coefficient:

$$
84 \times 8 = 672
$$

Thus, the coefficient of $x^6 y^3$ is 672.

Therefore, the correct answer is Option 1: 672.

**Explanation for Incorrect Options:**

- **Option 2 (572):** This option is incorrect because the calculation of the coefficient using the binomial theorem and the powers of 2 results in 672, not 572.
  
- **Option 3 (472):** This option is incorrect for the same reason as above. The correct calculation yields 672, not 472.
  
- **Option 4 (272):** This option is incorrect because it significantly underestimates the coefficient. The correct calculation, as shown, results in 672.

In conclusion, the correct coefficient of $x^6 y^3$ in the expansion of $(x + 2y)^9$ is indeed 672, which supports Option 1 as the correct answer.