The value of \( \sum_{\substack{r=0}}^{s} \sum_{\substack{s=1 \\ r \leq s}}^{n} C_{s}^{s} C_{r} ...
The value of \( \sum_{\substack{r=0}}^{s} \sum_{\substack{s=1 \\ r \leq s}}^{n} C_{s}^{s} C_{r} \) is
(a) \( 3^{n}-1 \)
(b) \( 3^{n}+1 \)
(c) \( 3^{n} \)
(d) \( 3\left(3^{n}-1\right) \)
📲PW App Link - https://bit.ly/YTAI_PWAP
🌐PW Website - https://www.pw.live
Other Videos By PW Solutions
2023-07-03 | Prove that \( 2222^{5555}+5555^{2222} \) is divisible by 7 . |
2023-07-03 | Prove that \( \sum_{m=0}^{n}{ }^{n} C_{m} \cdot \sin (m x) \cdot \cos (n-m) x=2^{n-1} \sin (n x) \) |
2023-07-03 | Let \( N_{1}=2^{55}+1 \) and \( N_{2}=165 \). Then sum of digits of the HCF of \( N_{1} \) and \... |
2023-07-03 | Match the columns:
\begin{tabular}{|l|l|l|l|}
\hline \multicolumn{2}{|c|}{ Column-I } & \multico... |
2023-07-03 | Sum of the series \( S=\sum_{\mathrm{r}=0}^{\mathrm{n}} \frac{3^{\mathrm{r}+4}\left({ }^{\mathrm... |
2023-07-03 | Let \( N=2^{1224}-1, \alpha=2^{153}+2^{77}+1 \) and \( \beta=2^{408}-2^{204}+1 \).
Then which of... |
2023-07-03 | Suppose \( [x] \) denote the greatest integer \( \leq \), and \( n \in N \), then \( \lim _{n \r... |
2023-07-03 | The remainder when \( 2^{30} \cdot 3^{20} \) is divided by 7 is: (a) 1 (b) 2 (c) 4 (d) 6 |
2023-07-03 | The coefficient of \( x^{83} \) in \( \left(1+x+x^{2}+x^{3}+x^{4}\right)^{n}(x-1)^{n+3} \) is eq... |
2023-07-03 | Suppose \( m \) and \( n \) are positive integers and let \( \mathrm{S}=\sum_{\mathrm{k}=0}^{\ma... |
2023-07-03 | The value of \( \sum_{\substack{r=0}}^{s} \sum_{\substack{s=1 \\ r \leq s}}^{n} C_{s}^{s} C_{r} ... |
2023-07-03 | The largest real value for \( x \) such that \( -\sum_{\mathrm{k}=0}^{4}\left(\frac{5^{4-\mathrm... |
2023-07-03 | Let \( S_{1}=\{(i, j, k): i, j, k \in\{1,2, \ldots, 10\}\} \)
\( S_{2}=\{(i, j): 1 \leq \mathrm{... |
2023-07-03 | Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls... |
2023-07-03 | Let \( x=(5 \sqrt{2}+7)^{19} \), then \( \mathrm{x}\{\mathrm{x}\}\{ \).\( \} .denotes the fracti... |
2023-07-03 | The number of natural numbers lying between 1012 and 23421 that can be formed using the digits \... |
2023-07-03 | The number of 4-digit integers in the closed interval \( [2022,4482] \) formed by using the digi... |
2023-07-03 | In the expansion of \( \left(1+3 x+2 x^{2}\right)^{6} \) the coefficient of \( x^{11} \) is
(a) ... |
2023-07-03 | How many different nine-digit numbers can be formed from the number 223355888 by rearranging its... |
2023-07-03 | Number of divisors of the form \( (4 n+2), n \geq 0 \) of the integer 240 is
(a) 4
(b) 8
(c) 10
... |
2023-07-03 | Let \( S \) be the set of all passwords which are six to eight characters long, where each chara... |