Binary sequences

Let $F(m,n,r,s)$ be the number of binary sequences containing:

• $m$ times the digit '0'
• $n$ times the digit '1'
• $r$ times the sequence '00'
• $s$ times the sequence '11'

For instance, $F(4,5,2,3)$ == 24 :

• 0 0 0 1 0 1 1 1 1
• 0 0 0 1 1 0 1 1 1
• 0 0 0 1 1 1 0 1 1
• 0 0 0 1 1 1 1 0 1
• 0 0 1 0 0 1 1 1 1
• 0 0 1 1 0 0 1 1 1
• 0 0 1 1 1 0 0 1 1
• 0 0 1 1 1 1 0 0 1
• 0 1 0 0 0 1 1 1 1
• 0 1 1 0 0 0 1 1 1
• 0 1 1 1 0 0 0 1 1
• 0 1 1 1 1 0 0 0 1
• 1 0 0 0 1 1 1 1 0
• 1 0 0 1 1 1 1 0 0
• 1 0 1 1 1 1 0 0 0
• 1 1 0 0 0 1 1 1 0
• 1 1 0 0 1 1 1 0 0
• 1 1 0 1 1 1 0 0 0
• 1 1 1 0 0 0 1 1 0
• 1 1 1 0 0 1 1 0 0
• 1 1 1 0 1 1 0 0 0
• 1 1 1 1 0 0 0 1 0
• 1 1 1 1 0 0 1 0 0
• 1 1 1 1 0 1 0 0 0

What is $F(777,555,333,111)$?

Answer format: the last 10 digits before the trailing 0 of this large number.

You are given 7741694224 for $F(77,55,33,11)$