Conductors need to be able to control and signal every part of a band. In order to do this they must have the full score so they know what each part of the band is playing. This allows them to indicate musical entries and keep each instrument in the band in time.
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The score shows all the various parts on a single page, so the conductor knows what everyone is supposed to be playing.
the conductor
A baton? His/her musicality and common sense?
The score is printed sheet music that shows all the different instruments' parts at the same time.
That stick orchestra conductors use is called a baton.