$ ext{If } (1 + 2x + x^2)^n = \sum_{r=0}^{2n} a_r x^r ext{, then }$ a_r =

Q: $\text{If } (1 + 2x + x^2)^n = \sum_{r=0}^{2n} a_r x^r \text{, then }$ a_r =

$\text{If } (1 + 2x + x^2)^n = \sum_{r=0}^{2n} a_r x^r \text{, then }$ a_r =

Question

$	ext{If } (1 + 2x + x^2)^n = \sum_{r=0}^{2n} a_r x^r 	ext{, then }$ a_r =

Accepted Answer

To determine the coefficient $ a_r $ in the expansion of $(1 + 2x + x^2)^n = \sum_{r=0}^{2n} a_r x^r$, we need to understand how the binomial expansion works in this context.

### Explanation for Option 1:

The expression $(1 + 2x + x^2)^n$ can be expanded using the multinomial theorem, which is a generalization of the binomial theorem. The multinomial theorem states that:

$$
(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2+\cdots+k_m=n} \frac{n!}{k_1!k_2!\cdots k_m!} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}
$$

In our case, $x_1 = 1$, $x_2 = 2x$, and $x_3 = x^2$. We are interested in the coefficient of $x^r$ in the expansion. The term $x^r$ can be formed by selecting terms from the expansion such that the powers of $x$ add up to $r$.

The coefficient of $x^r$ in the expansion is given by the sum of all possible ways to distribute the powers among the terms $1$, $2x$, and $x^2$ such that the total power of $x$ is $r$. This is equivalent to choosing $r$ objects from $2n$ possibilities, which is given by the binomial coefficient ${}^{2n}C_r$.

Thus, the correct answer is Option 1: ${}^{2n}C_r$.

### Explanation for why other options are incorrect:

- **Option 2: ${}^nC_r \cdot {}^nC_{r+1}$**

This option suggests a product of two binomial coefficients, which does not align with the multinomial expansion of $(1 + 2x + x^2)^n$. The structure of the problem does not involve a product of combinations from two separate binomial expansions, making this option incorrect.

- **Option 3: $\left( {}^nC_r \right)^2$**

This option implies squaring a single binomial coefficient, which is not relevant to the multinomial expansion of the given expression. The problem involves distributing powers among three terms, not squaring a single distribution, so this option is incorrect.

- **Option 4: ${}^{2n}C_{r+1}$**

This option suggests a binomial coefficient for $r+1$, which does not match the requirement for finding the coefficient of $x^r$. The correct coefficient should be based on choosing $r$ objects, not $r+1$, from $2n$ possibilities, making this option incorrect.

In conclusion, Option 1 is the correct choice because it accurately represents the coefficient of $x^r$ in the multinomial expansion of $(1 + 2x + x^2)^n$.