Question 22
Find the number of possible ways of arranging $m$ ones and $n$ zeros in a row such that there are in total $2k+1$ strings of ones and zeros. For example,
$$1110001001110001$$
consists of $4$ strings of ones and $3$ strings of zeros.
Editorial clarification: the paragraph above preserves the original scan. The supplied filename explicitly corrects this to $10$ ones, $11$ zeros and $13$ strings; Eytan Chong's supplementary compendium prints the same correction. For the numerical version assessed here, use $m=10$, $n=11$, $k=6$. A string means a maximal consecutive block of equal digits.
$$1110001001110001$$
consists of $4$ strings of ones and $3$ strings of zeros.
Editorial clarification: the paragraph above preserves the original scan. The supplied filename explicitly corrects this to $10$ ones, $11$ zeros and $13$ strings; Eytan Chong's supplementary compendium prints the same correction. For the numerical version assessed here, use $m=10$, $n=11$, $k=6$. A string means a maximal consecutive block of equal digits.