Where $A^{\prime}$ is the complement of $A$. Let $x$ be the number of integers lying between 2999 and 8001 which have at least two digits equal. Then x is equal to

Number of numbers between 2999 and 8001 $=8001-2999-1=5001$ \begin{array}{|l|l|l|l|}\hline 3 & & & \\\hline\end{array} Number of numbers with all digit distinct and having 3 as starting digit $=9 \times 8 \times 7=504$ Number of numbers with all digit distinct and having 4 as starting digit.
Similarly number of numbers with starting digit 5, 6 and 7 respectively are 504, 504 and 504. Total numbers $=5 \times 504=2520$ Hence, required number $=5001-2520=2481$