To solve the problem of finding the number of 4-digit numbers where the sum of digits is 10 and the number is divisible by 11, follow these steps:

Step 1: Define Variables and Divisibility Rule

Let the 4-digit number be (ABCD) ((A≠0)). For divisibility by 11:
((A + C) - (B + D) = 11k) (where (k) is an integer).

Let (S_1 = A + C) and (S_2 = B + D). We know:
(S_1 + S_2 = 10) (sum of digits is 10) and (\vert S_1 - S_2 \vert = 11k).

Step 2: Determine Valid (k)

Since (S_1) and (S_2) are non-negative integers, the only possible (k) is 0 (other multiples of 11 are too large/small). Thus:
(S_1 = S_2 = 5).

Step 3: Count Solutions for (A + C = 5)

(A≥1), so (A) can be 1-5 (since (C=5-A≥0)):
Number of solutions: 5.

Step 4: Count Solutions for (B + D = 5)

(B≥0), so (B) can be 0-5:
Number of solutions: 6.

Step 5: Total Number of Valid Numbers

Multiply the solutions: (5×6 = 30).

Answer: (\boxed{30})

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