10. Given a positive integer $n$ that satisfies the following conditions:
(1) It is an eight-digit number, and all its digits are 0 or 1;
(2) Its first digit is 1;
(3) The sum of the digits in the even positions equals the sum of the digits in the odd positions.
How many such $n$ are there?
10.35.

From the fact that the sum of the digits in the even positions equals the sum of the digits in the odd positions, we know that the number of 1s in the even positions equals the number of 1s in the odd positions.

Since the first digit is fixed, there are only three positions in the odd positions that can be freely changed.

Therefore, the total is $\sum_{k=1}^{4} \mathrm{C}_{3}^{k-1} \mathrm{C}_{4}^{k}=4+18+12+1=35$ possible integers.