The coefficient of \( x^{101} \) in the expression \( (5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+...

Channel:
PW Solutions
Subscribers:
443,000
Published on ● Video Link: https://www.youtube.com/watch?v=s1MmY8pp6Y4


Duration: 5:44
1 views
0

The coefficient of \( x^{101} \) in the expression \( (5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots+x^{500}, x0 \), is
(1) \( { }^{501} C_{101}(5)^{399} \)
(2) \( { }^{501} C_{101}(5)^{400} \)
(3) \( { }^{501} C_{100}(5)^{400} \)
(4) \( { }^{500} C_{101}(5)^{399} \)
📲PW App Link - https://bit.ly/YTAI_PWAP
🌐PW Website - https://www.pw.live



Other Videos By PW Solutions


2023-07-03Let \( C_{r} \) denote the binomial coefficient of \( x^{r} \) in the expansion of \( (1+x)^{10}...
2023-07-03If the coefficient of \( x^{10} \) in the binomial expansion of \( \left(\frac{\sqrt{x}}{5^{1 / ...
2023-07-03If \( \sum_{k=1}^{31}\left({ }^{31} C_{k}\right)\left({ }^{31} C_{k-1}\right)-\sum_{k=1}^{30}\le...
2023-07-03If \( \left({ }^{40} C_{0}\right)+\left({ }^{41} C_{1}\right)+\left({ }^{42} C_{2}\right)+\ldots...
2023-07-03The remainder when \( (2021)^{2022}+(2022)^{2021} \) is divided by 7 is (a) 0 (b) 1 (c) 2 (d) 6
2023-07-03The remainder when \( (2021)^{2023} \) is divided by 7 is (a) 1 (b) 2 (c) 5 (d) 6
2023-07-03Find the sum of following series: (i) \( \sum_{r=1}^{51} r\left({ }^{(1001-r)} C_{950}\right) \)...
2023-07-03If \( \sum_{r=0}^{n} \frac{2 r}{{ }^{n} C_{r}}=\sum_{r=0}^{n} \frac{n^{2}-3 n+3}{{ }^{n} C_{r}} ...
2023-07-03Find the sum of following series: (i) \( \sum_{r=0}^{49}{ }^{100} C_{r} \) (ii) \( \sum_{r=0}^{2...
2023-07-03Prove that \( \sum_{r=1}^{k}(-3)^{r-1} 3 n_{C_{2 r-1}}=0 \), where \( k=(3 n) / 2 \) and \( n \)...
2023-07-03The coefficient of \( x^{101} \) in the expression \( (5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+...
2023-07-03Find the sum of following series: (i) \( \sum_{r=0}^{n}{ }^{n} C_{r}(-1)^{r}\left(\frac{1}{2^{r}...
2023-07-03Prove That: \( \sqrt{C_{1}}+\sqrt{C_{2}}+\sqrt{C_{3}}+\ldots+\sqrt{C_{n}} \leq 2^{n-1}+\frac{n-1...
2023-07-03. If \( \sum_{r=0}^{2 n} a_{r}(x-2)^{r}=\sum_{r=0}^{2 n} b_{r}(x-3)^{r} \) and \( a_{k}=1 \) for...
2023-07-03If \( ^{n} J_{r_{r}}=\frac{\left(1-x^{n}\right)\left(1-x^{n-1}\right) \ldots\left(1-x^{n-r+1}\ri...
2023-07-03Show that \( \sum_{r=0}^{n} \frac{C_{r} \cdot 3^{r+4}}{(r+1)(r+2)(r+3)(r+4)} \) \[ -=\frac{1}{(n...
2023-07-03Prove that : \( { }^{\mathrm{n}} C_{0} \cdot{ }^{2 n} C_{n}-{ }^{n} C_{1} \cdot{ }^{2 n-2} C_{n}...
2023-07-03Find the sum of possible real Values of \( x \) for which the sixth term of the expansion of \( ...
2023-07-03Match the columns: \begin{tabular}{|c|l|l|l|} \hline \multicolumn{2}{|c|}{ Column-I } & \multico...
2023-07-03Find the number of divisors of the number \[ \mathrm{N}={ }^{2000} C_{1}+2 \cdot{ }^{2000} C_{2}...
2023-07-03Real value for \( x \) such that \( \sum_{k=0}^{4}\left(\frac{3^{4-k}}{(4-k) !}\right)\left(\fra...