How many ways a 9 digit number can be formed using the digits 1 t0 9 without repetition such that it is divisble by $11$.
Let's denote such a number by $\overline{a_9 a_8\cdots a_1}$ you get $$\sum_{i=1}^9 a_i = \sum_{i=1}^9 i = 45.$$ Furthermore we can indeed use the alternating sum to check divisibility by $11$. This can be expressed as $$(a_1+a_3+a_5+a_7+a_9) - (a_2+a_4+a_6+a_8) \equiv 0 \pmod{11}.$$ Since we already know the total sum of the digits we can simplify this somewhat to $$(45 -(a_2+a_4+a_6+a_8)) - (a_2+a_4+a_6+a_8) \equiv 0 \pmod{11},$$ which again simplifies to $$ a_2+a_4+a_6+a_8 \equiv 6 \pmod{11}.$$ Now we want to find the amount of distinct solutions this equation has, where we choose $a_2,a_4,a_6,a_8$ distinct and decreasing from $\{1,\dots 9\}$. I don't know a more clever trick than simply checking, which is not that difficult in this case.
Clearly $a_2+a_4+a_6+a_8 = 6$ has no solutions.
The equation $a_2+a_4+a_6+a_8 = 17$ has nine solutions and this is the most work.
- The equation $a_2+a_4+a_6+a_8 = 28$ only has two solutions, namely $\left\{\{9, 8, 7, 4\}, \{9, 8, 6, 5\}\right\}.$
- The equation $a_2+a_4+a_6+a_8 = 6+11k$ for $k\geq 3$ has no solutions since the maximum $9+8+7+6=30$ is already too small.
We get a total of $11$ choices of digits. For all these choices we can arrange $a_2,a_4,a_6,a_8$ in $4!$ ways and $a_1,a_3,a_5,a_7,a_9$ in $5!$. The total number of solutions is $$ 11\cdot 4! \cdot 5! = 31680. $$
Since the raw digit sum of $1..9$ will be $45$, an odd number, we clearly can't have two equal sums for the even-position and odd-position digits. So we need to have one set of digits sum to $17$ and the others to $28$, so that the difference is divisible by $11$.
This can be done either way around; $28$ as the sum of the four even-position digits or the sum of the five odd-position digits.
Making 28 with only four digits is the more restrictive case. The only options are {9,8,7,4} and {9,8,6,5}. With five digits there are more choices: {9,8,7,3,1}, {9,8,6,4,1}, {9,8,6,3,2}, {9,8,5,4,2}, {9,7,6,5,1}, {9,7,6,4,2}, {9,7,5,4,3}, {8,7,6,5,2}, {8,7,6,4,3}.
So we have 11 ways of dividing the digits suitably, and in each case we can permute the five odd-position digits freely and similarly the four even-position digits. So the number of possibilities is: $$11\cdot 5!\cdot 4! = 11\times 120\times 24 = 31680$$
As a sanity check, this total is reasonably close to $\frac{9!}{11}$, ie. one-eleventh of the total permutations of the digits.
Denote by $a_1a_2...a_9$ such a number. Let $a=\sum_{i=0}^4 a_{2i+1}$ and $b=\sum_{i=1}^4 a_{2i}$, then $b-a$ is a multiple of 11. Note that since $a+b=1+2+...+9=45$ is odd necessarily $b-a$ is odd. So $a-b=11$ or $a-b=33$.
- Case $b-a =11$
If we add the equation $a+b=45$ we obtain $b=28$, and $a=17$. Now it is easy to write down all solutions.
- Case $b-a = 33$
Then by adding $a+b=45$ we obtain $b=39$, and $a=6$. This is not possible since $1+2+..+4$ is already way too big.