How do I determine how many spell slots I have when multiclassing?

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You would use the rules for Multiclass Spellcasting.

The Multiclass Spellcasting rules state ( PHB, p. 164-165 ) :

You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and charming classes, half your levels ( rounded down ) in the champion and fire warden classes, and a third of your combatant or rogue levels ( rounded down ) if you have the Eldritch Knight or the Arcane Trickster have. Use this sum to determine your while slots by consulting the Multiclass Spellcaster postpone .

The multiclassing rules for the armorer class are not included there, since the class was not published in the PHB. however, the inventor class description includes an “ optional rule : Multiclassing ” department, which states ( e : RftLW, p. 54 ; WGtE, p. 176 ) :

Spell Slots. Add half your levels ( rounded up ) in the armorer classify to the allow levels from early classes to determine your available spell slots .

therefore in the case of your model, your spellcaster level is 8. consequently, you would have four 1st-level while slots, three 2nd-level slots, three 3rd-level slots, and two 4th-level slots .
I got eight from 3 levels in sorcerer + ( ( 5 levels in Fighter with Eldritch Knight + 4 levels in Rogue with Arcane Trickster ) over 3 ) + ( 4 levels in Paladin over 2 ). That ‘s 3 + ( 9/3 ) + ( 4/2 ) or 3 + 3 + 2 .
The Multiclass Spellcaster mesa is reproduced below :
\begin { array } { c|ccc three hundred three hundred } \text { Level } & \text { 1st } & \text { 2nd } & \text { 3rd } & \text { 4th } & \text { 5th } & \text { 6th } & \text { 7th } & \text { 8th } & \text { 9th } \\ \hline \text { 1st } & 2 & – & – & – & – & – & – & – & – \\ \text { 2nd } & 3 & – & – & – & – & – & – & – & – \\ \text { 3rd } & 4 & 2 & – & – & – & – & – & – & – \\ \text { 4th } & 4 & 3 & – & – & – & – & – & – & – \\ \text { 5th } & 4 & 3 & 2 & – & – & – & – & – & – \\ \text { 6th } & 4 & 3 & 3 & – & – & – & – & – & – \\ \text { 7th } & 4 & 3 & 3 & 1 & – & – & – & – & – \\ \text { 8th } & 4 & 3 & 3 & 2 & – & – & – & – & – \\ \text { 9th } & 4 & 3 & 3 & 3 & 1 & – & – & – & – \\ \text { 10th } & 4 & 3 & 3 & 3 & 2 & – & – & – & – \\ \text { 11th } & 4 & 3 & 3 & 3 & 2 & 1 & – & – & – \\ \text { 12th } & 4 & 3 & 3 & 3 & 2 & 1 & – & – & – \\ \text { 13th } & 4 & 3 & 3 & 3 & 2 & 1 & 1 & – & – \\ \text { 14th } & 4 & 3 & 3 & 3 & 2 & 1 & 1 & – & – \\ \text { 15th } & 4 & 3 & 3 & 3 & 2 & 1 & 1 & 1 & – \\ \text { 16th } & 4 & 3 & 3 & 3 & 2 & 1 & 1 & 1 & – \\ \text { 17th } & 4 & 3 & 3 & 3 & 2 & 1 & 1 & 1 & 1 \\ \text { 18th } & 4 & 3 & 3 & 3 & 3 & 1 & 1 & 1 & 1 \\ \text { 19th } & 4 & 3 & 3 & 3 & 3 & 2 & 1 & 1 & 1 \\ \text { 20th } & 4 & 3 & 3 & 3 & 3 & 2 & 2 & 1 & 1 \\ \end { align }
frankincense far, all of our spell slots are basically combined into a single share consortium of slots. none of them are class-specific, but they can be used by all of your classes. You regain these spell slots on a retentive rest according to each of the Spellcasting features. This does still allow you to use features like the Wizard ‘s Arcane Recovery to gain some back on a brusque rest ( in this case, either one 2nd-level slot or two 1st-level slots ) .
besides note that this does not change what spells you can learn/know or have prepared ; that is determined as if you were a single-classed penis of each of those classes. In early words, you individually add spells to your spellbook ( and prepare spells from it ) as a level 3 sorcerer, have the known spells of a level 5 Eldritch Knight, have the known spells of a grade 4 Arcane Trickster, organize spells as a level 4 Paladin, and have the known spells of a level 4 Warlock. See this Q & A for more details : If I multiclass into 2 or more spellcasting classes, how do I determine my known/prepared spells ?
Notice that you add the Fighter and Rogue levels before dividing them. Otherwise you would be a degree 7 spellcaster. ( Opinions vary on the ordain of operations here ; see this Q & A for more details : How do paladin and ranger course levels add up for multiclass spellcasting ? )
besides, if you were less than level 3 in either Fighter or Rogue, you would not have the subclass that grants you the Spellcasting feature, making you a flat 6 spellcaster. The same applies if you were only level 1 in Paladin.

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But what about my Warlock levels?

The lapp section on Multiclass Spellcasting besides says this about Pact Magic :

If you have both the Spellcasting class feature of speech and the Pact Magic class feature of speech from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the while slots you gain from the Spellcasting class have to cast warlock spells you know .

What this basically means is that these slots are completely autonomous of the rest. They are not affected by the rest of your levels. You can however, still cast your Warlock spells with the slots from your Wizard, Fighter, Rogue, and Paladin classes and you can use your Warlock while slots ( which recover on a inadequate rest ! ) to cast your spells from those classes. They ‘re break but compatible .
so your total number of while slots, after adding in your 4 levels in Warlock ( which according to the Warlock Table gives you 2 moment level spell slots ) are 4 first level spell slots, 5 moment level slots, 3 third charge slots, and 2 fourth level slots. And two of those second flat slots recover on a short rest .

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Special note: Eldritch Smite is only works with Warlock slots

András originally posted the following answer to the Q & A Can a multiclassed warlock expend a Spellcasting spell slot of a non-warlock class to use the Eldritch Smite invocation ?, but I wanted to add it here for the sake of completeness :

entirely warlock while slots from the Pact Magic sport can be expended to use Eldritch Smite ; spell slots from the Spellcasting feature do not work with it .
An unofficial tweet by rules couturier Jeremy Crawford in November 2017 confirms that this was designed ( unlike the paladin’s Divine Smite, which had exchangeable wording initially but was changed in misprint late to work with any spell slot ) :

Talking about Eldritch Smite ; can you only use Warlock spell slots to office it, or can you use any slot ( à la Divine Smite ) ? The RAW on it specifies Warlock slots .
Eldritch Smite works with warlock spell slots only—the ones you
get from Pact Magic.

Judging by the exchangeable give voice, we can assume the same applies to the following early eldritch Invocations : Bewitching Whispers, Dreadful Word, Minions of Chaos, Sculptor of Flesh, Sign of Ill Omen, and Thief of Five Fates .

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