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I am currently playing in a 5e game as a multiclassed Arcane Trickster 10 / Alchemist 3. According to the Artificer multiclassing rules(TCE pg. 10) and the general multiclassing rules(PHB pgs. 164 and 165) I am a 5th level caster with four 1st level spell slots, three 2nd level slots and two 3rd level slots. (Arcane Trickster 10/3 rounded down = 3. Alchemist 3/2 rounded up = 2. 3+2=5)
The Spells Known and Prepared section of the multiclassing rules(PHB pg. 164) says to prepare spells for each class separately. Since neither an Arcane Trickster 10, nor an Alchemist 3 would have 3rd level spells, does this mean that I cannot prepare 3rd level spells, even though I have the slots for it?