A-matrix-has-N-rows-and-2k-1-columns-Each-column-is-filled-with-M-ones-and-N-M-zeros-A-given-row-j-is-cool-if-and-only-if-i-1-2k-1-a-ji-k-Find-the-minimum-and-the-maximum-number-of-c

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Question Number 19547 by dioph last updated on 12/Aug/17

$$\mathrm{A}\mathrm{matrix}\mathrm{has}N\mathrm{rows}\mathrm{and}2k-1$$$$\mathrm{columns}.\mathrm{Each}\mathrm{column}\mathrm{is}\mathrm{filled}\mathrm{with}$$$$M\mathrm{ones}\mathrm{and}N-M\mathrm{zeros}.$$$$\mathrm{A}\mathrm{given}\mathrm{row}j\mathrm{is}“{cool}^{″}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}$$$$\underset{i=1}{\sum ^{2k-1}}{a}_{ji}⩾k.\mathrm{Find}\mathrm{the}\mathrm{minimum}\mathrm{and}$$$$\mathrm{the}\mathrm{maximum}\mathrm{number}\mathrm{of}\mathrm{cool}\mathrm{rows}$$$$\mathrm{for}\mathrm{given}N,k\mathrm{and}M.$$

Commented by dioph last updated on 12/Aug/17

$$\mathrm{For}\mathrm{instance},\mathrm{if}N=5,k=2,M=2$$$$\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]\mathrm{has}1\mathrm{cool}\mathrm{row}\left(\min \right)$$$$\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 0& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\mathrm{has}3\mathrm{cool}\mathrm{rows}\left(\max \right)$$

Commented by dioph last updated on 31/Aug/17

$$\mathrm{anyone}?$$

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