22. Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more than three 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition.
22. Ans: 24068
Consider the sequence of ten I's. There are eleven spaces between two 1's, or before the leftmost 1 or after the rightmost 1. For each of these spaces, we can put either two 0 's (double 0 's) or only one 0 (single 0 ). If there are exactly $k$ doubles 0 's, then there are only $10-2 k$ single 0 's. The number of ways for this to happen is $\binom{11}{k}\binom{11-k}{10-2 k}$. Thus the answer is
$$
\sum_{k=0}^{5}\binom{11}{k}\binom{11-k}{10-2 k}=24068 .
$$