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# Counting

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Can anyone help plz?

A debate team consists of 5 freshmen and 4 sophomores.  (Everyone is distinguishable.)  In how many ways can they stand in line, so at least two of the sophomores are standing next to each other, and at least two of the freshmen are at the ends of the line?

Feb 22, 2023

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If 2 freshmen are at both ends, there are $$5 \times 4 \times 3 \times 2 = 60$$ ways to order the freshman.

Then, there are $$5!$$ ways to order the rest (note that there will always be 2 sophomores next to each other).

So, there are $$5! \times 60 = \color{brown}\boxed{7200}$$ ways to order the students.

Feb 22, 2023