25. Find the number of $0-1$ binary sequences formed by six 0 's and six 1 's such that no three 0's are together. For example, 110010100101 is such a sequence but 101011000101 and 110101100001 are not.
25. Answer: 357
First put the six 1 's in one sequence. Then there are 7 gaps before the first 1 , between two adjacent 1 's and after the last 1 . For each such gap, we can put a single 0 or double 0 's (that is, 00 ).
If there are exactly $i$ double 0 's, then there are exactly $6-2 i$ single 0 's, where $i=$ $0,1,2,3$. Therefore the number of such binary sequences with exactly $i$ double 0 's is $\binom{7}{i}\binom{7-i}{6-2 i}$. Hence the answers is $\sum_{i=0}^{3}\binom{7}{i}\binom{7-i}{6-2 i}=357$.