Permutations & Combinations · JEE Advanced previous year questions

Let $S$ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $S$, but 0210222 is $\text{NOT}$ in $S$. Then the number of elements $x$ in $S$ such that at least one of the digits 0 and 1 appears exactly twice in $x$, is equal to ____.

A group of $9$ students, $s_1,s_2,\dots ,s_9$, is to be divided to form three teams $X,Y$, and $Z$ of sizes $2,3$, and $4$, respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is _____.

Let $7\stackrel{r}{\overbrace{5\cdots 5}}7$ denote the $(r+2)$ digit number where the first and the last digits are 7 and the remaining $r$ digits are 5. Consider the sum $S=77+757+7557+\cdots +7\stackrel{98}{\overbrace{5\cdots 5}}7$. If $S=\frac{7\stackrel{99}{\overbrace{5\cdots 5}}7+m}{n}$, where $m$ and $n$ are natural numbers less than 3000, then the value of $m+n$ is ____.