Note that monks and barbarians aren't compatible due to alignment restrictions. There's a PFS FAQ about changing alignment, I think.
Alignment aside.
Bab: +7/+2 (monk), +1 (barb), +0 (rogue) = +8/+3.
Flurry: +10/+5 (monk), +1 (barb), +0 (rogue) = +11/+6.
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Flurry of Blows (Ex): Starting at 1st level, a monk can make a flurry of blows as a full-attack action. When doing so he may make one additional attack using any combination of unarmed strikes or attacks with a special monk weapon (kama, nunchaku, quarterstaff, sai, shuriken, and siangham) as if using the Two-Weapon Fighting feat (even if the monk does not meet the prerequisites for the feat). For the purpose of these attacks, the monk's base attack bonus from his monk class levels. For all other purposes, such as qualifying for a feat or a prestige class, the monk uses his normal base attack bonus.
At 8th level, the monk can make two additional attacks when he uses flurry of blows, as if using Improved Two-Weapon Fighting (even if the monk does not meet the prerequisites for the feat).
At 15th level, the monk can make three additional attacks using flurry of blows, as if using Greater Two-Weapon Fighting (even if the monk does not meet the prerequisites for the feat).
Monk: The monk rules for flurry state, "For the purpose of these attacks, the monk's base attack bonus is equal to his monk level." How does this interact with BAB from class levels and racial Hit Dice? Does a multiclassed fighter 19/monk 1 flurry as if his BAB were only +1?
A monk using flurry treats his BAB from monk levels as equal to his monk level. He still adds BAB from other sources (such as other classes or racial Hit Dice) normally to this total.
So a fighter 19/monk 1 has a normal BAB of +19. When he flurries, he treats his monk BAB as +1 (for his 1 level of monk) and still gets BAB +19 from his fighter levels, for a total flurry BAB of +20.
-- Calculating BAB is "Like, duh"; nobody needs FAQ help to add that up. What we need to know is everything else.
I.e., a 7th level monk (BAB5, Flurry +5/+5/+0) who advances to monk8 gains the class text-stated equivalent of Flurry-granted Improved Two Weapon Fighting (for Flurry +6/+6/+1/+1).
But suppose he takes a level a fighter, becoming fighter1/monk7 with a BAB of 6....
Right now, my understanding is that a F1/M7 does not gain an extra attack since he is not a real monk8 and therefore has not yet attained Flurry-cum-Improved Two Weapon Fighting as a class feature. He Flurries at +6/+6/+1, which is an improvement over the +4/+4/-1 a BAB6 non-monk PC using TWF (but not ITWF) would receive, but no additional attack is being granted.
IOW, you can't just add up monk levels and non-monk BAB and then zoom to that level-number on the monk chart to look up Flurry attack number and bonus.
(Everyone please jackhammer the FAQ button on this post. Thankyaverymuch.)
There's nothing to FAQ.
Your number of flurry's is based on your MONK LEVEL, not your Monk BAB. That's why it's a row on the table. The only modifier that is affected by the multiclassing is the BAB (which is addressed by the FAQ). A monk 1/Fighter 19 has a flurry of 2 blows, both of which use his 20 BAB as a base (-2 for flurrying).
Find monk level, read across to get number of attacks. Adjust listed BAB to be monk level (the BAB's on the line) plus the multiclass BAB bonuses. Done.
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There's nothing to FAQ.
Your number of flurry's is based on your MONK LEVEL, not your Monk BAB. That's why it's a row on the table. The only modifier that is affected by the multiclassing is the BAB (which is addressed by the FAQ). A monk 1/Fighter 19 has a flurry of 2 blows, both of which use his 20 BAB as a base (-2 for flurrying).
Find monk level, read across to get number of attacks. Adjust listed BAB to be monk level (the BAB's on the line) plus the multiclass BAB bonuses. Done.
Until a FAQ is explicit in the matter, there are vast, heaping gobs of players out there who'll continue to get it wrong. I.e., a lot of F1/M7s are under the impression they get four Flurry attacks because they have BAB6.
The weird cases are all the levels just before the class grants an extra attack, not cream-puff examples like monk1/fighter19.
I don't think it really needs a FAQ, but I'll FAQ it for you.
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I'm looking at Martial artist monk 10, 1 rogue, 1 Barbarian(Original post for context.)
My understanding is the flurry there would be: +11/+11/+6/+6/+1Alignment aside.
Bab: +7/+2 (monk), +1 (barb), +0 (rogue) = +8/+3.
Flurry: +10/+5 (monk), +1 (barb), +0 (rogue) = +11/+6.
You're not subtracting two from each attack (for it being a Flurry); a Monk11 Flurries for +9/+9/+4/+4/–1.
...in this particular example, a Monk10/Barb1 has identical Flurry to Monk11 because the build is not at the cusp of any break point whereby the straight-class monk would pick up an extra attack as a class feature (as they do at 8th and 15th) whereas the multiclass monk wouldn't (see my previous posts concerning a Monk7/Figh1). A straight class monk's number of attacks increase by one each level from 14th to 16th (five to seven); and players of multiclasses whose BAB+(monk levels) add up to 15+ often think they're entitled to more attacks than they should be getting.
It's pretty blatant that a figh19/monk1 would be doing something wrong if he Flurried as if he had picked up Greater Two Weapon Fighting; it's less obvious in cases such as figh2/monk6 or barb2/figh4/[martial-artist]12.
Monk: Does a Monk 7 who multiclasses fighter receive an extra Flurry attack due to reaching BAB 6?"
No. Flurry grants additional attacks "at 8th [and 15th] level", not "at BAB +6 [and +11]." Yes, for the monk, BAB +6 coincides with 8th level. 8th level also brings the monk saving throws of +6, an increase in unarmed damage die size, and improved AC bonus due to level. Like all those other things, the BAB isn't mentioned and therefore isn't the determining factor; the monk's class level is.
As someone currently running a Monk(Zen Archer) 2/Fighter(Archer) 2/Ranger 5, believe me I wish it were otherwise.