The total number of three-digit numbers, with one digit repeated exactly two times, is

An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1$-15$ June $2021$ is. . . . . . 

$^{n - 1}{C_r} = ({k^2} - 3)\,.{\,^n}{C_{r + 1}}$ if $k \in $

Let $S=\{1,2,3,5,7,10,11\}$. The number of nonempty subsets of $S$ that have the sum of all elements a multiple of $3$ , is $........$

The value of $\sum \limits_{ r =0}^{20}{ }^{50- r } C _{6}$ is equal to

If $\sum\limits_{i = 0}^4 {^{4 + 1}} {C_i} + \sum\limits_{j = 6}^9 {^{3 + j}} {C_j} = {\,^x}{C_y}$ ($x$ is prime number), then which one of the following is incorrect