I think that the above answer is a gross over count. Although part of the analysis is right on.

Any 16-dight palindrome is completely determined by the first eight digits.

Because 16=(4)(4)=(8+8), it is clear that the sum must be 8 and the product of the non-zero digits must be 4.

Therefore as was found above, there are two possiblies: 41111000 or 22111100.

So the answer is the sum of the number of ways to rearrange each of those strings that do not begin with a zero.

Rearrangements of the string "41111000" can begin with 4 in

ways and can begin with a 1 in

.

Rearrangements of the string "22111100" can begin with a 2 in

ways and can begin with a 1 in

.

What is that sum?