1. Divisibility rule for 11

For a 3-digit number $abc$, the number is divisible by $11$ if $a-b+c=0,\ \pm 11.$ Since $a,b,c$ are digits and the number is at most $500$, we will list valid numbers directly.

2. Allowed digits

• The number is 3-digit, so the hundreds digit $a\neq 0$.
• Digit $1$ is not allowed.
• Number must be $\le 500$, so $a\in\{2,3,4\}$.
• Other digits can be from $\{0,2,3,4,5,6,7,8,9\}$.

3. Form of 3-digit multiples of 11

A 3-digit multiple of $11$ has the form: $110,121,132,143,154,165,176,187,198,209,220,231,242,253,264,275,286,297,308,319,330,341,352,363,374,385,396,407,418,429,440,451,462,473,484,495$ up to $500$.

4. Remove numbers containing digit $1$

From the above list, exclude all numbers having digit $1$: $121,143,154,165,176,187,198,319,341,418,451$ Also $110$ contains digit $1$, so exclude it too.

Thus the valid numbers are: $209,220,231,242,253,264,275,286,297,308,330,352,363,374,385,396,407,429,440,462,473,484,495$

5. Add them

Let us sum carefully: [ \begin{aligned} &209+220+231+242+253+264+275+286+297 \ &= 2277 \end{aligned} ]

[ \begin{aligned} &308+330+352+363+374+385+396+407+429+440+462+473+484+495 \ &= 5698 \end{aligned} ]

So total sum is $2277+5698=7975.$

6. Final answer

The required sum is $\boxed{7975}.$

7. Comparison with stored answer

Stored correct answer is $7744$, but the correct computed sum is $7975$. Hence, I do not agree with the stored answer.