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It has already been shown here that the booming blade cantrip and the Spell Sniper feat do work together, and here that the Sorcerer's Distant Spell Metamagic option and Spell Sniper also stack.

I am wondering if there is any reason the following build does not work:

Alice is a Bugbear multiclassed Sorcerer/Fighter (Battle Master) with the Spell Sniper feat, and she casts booming blade.

The range of booming blade is usually 5 feet. However, she has Spell Sniper which doubles the range to 10 feet, and she uses Distant Spell, which doubles the range again, to 20 feet.

She is holding a reach weapon, which has a reach of 10 feet. And she uses the Battle Master maneuver Lunging Attack, which increases the reach to 15. She then benefits from the Bugbear's Long-Limbed Feature, which increases the reach to 20.

Thus booming blade has a range of 20 feet and Alice has a reach of 20 feet, and so she can use booming blade from 20 feet away.

Is this correct? Have I correctly interpreted how these features work together to allow booming blade to work from 20 feet away?

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Yes

Battlemaster's Lunging Strike requires a melee weapon attack:

When you make a melee weapon attack on your turn, you can expend one superiority die to increase your reach for that attack by 5 feet.

Usually, a spell would involve a melee spell attack (things like Shocking Grasp or Vampiric Touch, for example). Booming Blade is an odd duck in that in that it directs the caster to make a melee weapon attack as part of casting the spell:

As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell's range.

This means that you are permitted to use Lunging Strike on Booming Blade.

The matter of stacking reach is pretty straightforward.

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