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From page 164 of the PHB (here is the equivalent section of the Basic Rules):

Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.

If I have a level 3 ranger and multiclass into paladin, how are my available spell slots calculated?

  • (ranger/2, rounded down) plus (paladin/2, rounded down)
  • (ranger + paladin)/2, rounded down

Which one is it?

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Your second way.

"... by adding together all your levels" would mean that your second example (ranger + paladin)/2 rounded down is how you'd work it. The rule doesn't have you segregate the levels before dividing if they are the same category. The example given in the book is a case of two different categories being combined, otherwise it would not have arrived at 5th level.

In your example, the result is 2 on the table, giving you 3 first level spells total for both this level and the next level you attain (be it 3/2 or 4/1).

The categories boil down to four the way the rules are presented:

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    \$\begingroup\$ It's worth noting that if you only round after you've done all the other math that the result is the same. I had an answer written based on this but figured it didn't add anything on it's own to the question. \$\endgroup\$ – LegendaryDude May 2 '16 at 13:00
  • \$\begingroup\$ @LegendaryDude - really? I thought if the levels were both odd then it wasn't the same, hence asking \$\endgroup\$ – Beornwulf May 2 '16 at 13:01
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    \$\begingroup\$ @Beornwulf like I said, if you only round the final result and not after each division, the result is the same. It's easy enough to test this with a few different cases to see that it is true. In your example, Ranger 3 Paladin 1, 1.5 + .5 = 2, 2 rounds to 2. \$\endgroup\$ – LegendaryDude May 2 '16 at 13:03
  • \$\begingroup\$ @LegendaryDude Ah, right, yes, I understand what you mean now \$\endgroup\$ – Beornwulf May 2 '16 at 13:06
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    \$\begingroup\$ @LegendaryDude That's not always the case though. Ranger 3 and Paladin 3 would be 2 if you rounded them down individually, but divide and add them after rounding you get 3. \$\endgroup\$ – Piomicron Jun 2 '18 at 23:25
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Your first way is correct

According to Jeremy Crawford's tweet here:

Multiclass spell slots: when dividing the levels of multiple classes, you divide, round down, and then add the results together.

Also remember Paladin and Ranger don't have the spell casting feature until 2nd level so you can't add them until you have at least 2 levels in either class. Which is another indicator that you round down before adding.

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As much as I am loathe to say it in any answer...

Consult your DM

The rules on what order to add levels together, when to divide, and when to round are unclear. The normal rules of mathematical operations can't help us because it's not actually presented as an equation. The language used is imprecise - it doesn't use any words that indicate order of operations or grouping (such as "then" or "combined").

Unfortunately, the guy who's supposed to provide clarity refused to do so:

Q: RAI, it seems multiclassing shouldn't be terrible. RAW, due to rounding, an EK5/AT5 doesn't have 2nd lvl spells. Thoughts?

A: Multiclassing—this optional rule is designed to make class combinations possible, not necessarily ideal. Work with your DM to fine tune.

Jeremy Crawford, via Twitter

When told the question was being asked from an AL standpoint, where "DM discretion" can't be consistently applied, he declined to provide any further response.

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