" To me, you need the "in which" because what is meant by this sentence is that the particular month is either unknown, of little importance, or applies to any or every month. The example, "My first final starts on the first day of the next month. " is not the same because you are specifying "next month" , thus, the month is known.
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jooneyAll promotions now begin on the first day of the month in which they are scheduled to run and end on the last day of that same month.Yes. in the month. The promotion is scheduled to "run in a certain month" - not scheduled to "run that month". You "run a promotion", not "run a month".
Do we need the preposition in?