For how many different ways, can you place 14 students in two rows of desks, where 5 of them sit always in first row, and 4 of them sit always in second row? Eight students can sit in on row.
2026-04-07 02:08:31.1775527711
For how many different ways, can you place 14 students in two rows of desks?
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Assuming a maximum of 8 students can sit in one row, 5-4 are fixed for first/second rows, there are 5 "floaters", and that all 14 need to be seated,
possible configurations are $(5+3)-(4+2),\; (5+2)-(4+3),\; (5+1)-(4+4)$
If you choose the "floaters" for the first row, those for the 2nd row automatically get decided,
and then you permute for each row, thus
$\binom{5}{3}*^8P_8*^8P_6 + \binom{5}{2}*^8P_7*^8P_7 + \binom{5}{1}*^8P_6*^8P_8$