2. Determine the number of those 0-1 binary sequences of ten 0 's and ten 1 's which do not contain three 0's together.
2. Ans: 24068
In such a binary sequence, 0 's either appear singly or in blocks of 2 . If the sequence has exactly $m$ blocks of double 0 's, then there are $10-2 m$ single 0 's. The number of such binary sequences is
$$
\binom{11}{m} \times\binom{ 11-m}{10-2 m}
$$

Thus the answer is
$$
\sum_{m=0}^{5}\binom{11}{m} \times\binom{ 11-m}{10-2 m}=24068
$$