Question:

Step 1: To find the coefficient of $x^{301}$ in the given expression $(1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\cdots+x^{500}$, we first need to understand the pattern of the terms and how they contribute to the $x^{301}$ term. The expression can be seen as a sum of terms where each term is of the form $x^k(1+x)^{500-k}$. Step 2: We can use the Binomial Theorem to expand each term. The Binomial Theorem states that for any positive integer $n$, $(a+b)^n = \sum_{k=0}^{n} {}^{n}C_{k}a^{n-k}b^k$. Applying this to our term, $x^k(1+x)^{500-k}$, we get $x^k \cdot \sum_{j=0}^{500-k} {}^{500-k}C_{j}x^j$. This means each term in the original expression contributes to the coefficient of $x^{301}$ when $k+j=301$. Step 3: To find the coefficient of $x^{301}$, we consider the contribution of each term $x^k(1+x)^{500-k}$ to $x^{301}$. This happens when $k+j=301$, where $j$ is the power of $x$ in the expansion of $(1+x)^{500-k}$. The coefficient of $x^{301}$ in the entire expression is thus the sum of coefficients of $x^{301}$ from each term, which can be found using the binomial coefficients. Step 4: The given expression can be re-written as $\sum_{k=0}^{500} x^k(1+x)^{500-k}$. This resembles the form of $(1+x)^{501}$ when expanded using the Binomial Theorem, which is $\sum_{k=0}^{501} {}^{501}C_{k}x^k$. Recognizing this pattern, we see that our expression is equivalent to $(1+x)^{501}$, but without the $x^{501}$ term, implying it's equivalent to $(1+x)^{501} - x^{501}$. Step 5: The coefficient of $x^{301}$ in $(1+x)^{501}$ can be directly found using the binomial coefficient formula, which is ${}^{n}C_{k} = \frac{n!}{k!(n-k)!}$. For $(1+x)^{501}$, the coefficient of $x^{301}$ is ${}^{501}C_{301}$. However, considering the relationship between the given series and the expansion of $(1+x)^{501}$, and noting that we are looking for the coefficient of $x^{301}$ which would be the same as the coefficient of $x^{200}$ in $(1+x)^{501}$ due to symmetry in binomial coefficients, we find that the coefficient we are looking for is actually ${}^{501}C_{200}$. Step 6: Therefore, by understanding the pattern of the given expression and relating it to the binomial expansion of $(1+x)^{501}$, we conclude that the coefficient of $x^{301}$ in the given expression is indeed ${}^{501}C_{200}$. This matches Option 4, confirming that the correct answer is ${}^{501}C_{200}$. The final answer is: $\boxed{4}$