11. How many four-digit numbers greater than 5000 can be formed from the digits $0,1,2$. $3,4,5,6,7,8,9$ if only the digit 4 may be repeated?
11. Answer: 2645.
Let $\overline{a b c d}$ represent the integer $a \times 10^{3}+b \times 10^{2}+c \times 10+d$.
Note that $\overline{a b c d}>5000$ iff $a \geq 5$ and $b, c, d$ are not all 0 . a must be a number in $\{5,6,7,8,9\}$. Suppose that $a$ is selected from $\{5,6,7,8,9\}$.
If 4 is not repeated, then the number of ways to choose $b, c, d$ is $9 \times 8 \times 7$;
If 4 appears exactly twice, then the number of ways to choose $b, c, d$ is $\binom{3}{2} \times 8$;
If 4 appears exactly three times, then the number of ways to choose $b, c, d$ is 1 .
Hence the answer is
$$
5 \times\left(9 \times 8 \times 7+\binom{3}{2} \times 8+1\right)=2645
$$