Hello, boxermc!

Here's part of it . . . hope it helps.

There is a 19 digit code that consists of 1's and 0's and starts with 1.

How many 19-digit codes have five more 0's than 1's or five more 1's than 0's?

If there are five more 0's than 1's, there must be seven 1's and twelve 0's.

One of the 1's is placed in the first slot.

The other 18 digits can be placed in ways.

If there are five more 1's than 0's, there must be twelve 1's and seven 0's.

One of the 1's is placed in the first slot.

The other 18 digits can be placed in ways.

Therefore, there are: . ways.