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EDIT: Thank you for all the links you provided answering the first part of the post, regarding the stacking of Polearm Master (PAM), War Caster (WC) work and Spell Sniper (SS) feats. [if someone didn't know, the consensus is that combining the three will allow you to cast booming blade in place of an opportunity attack when an enemy enters or exits your reach of 10 ft]

I would be grateful if someone could help with the second part regarding the damage output.
From a damage point of view, using the cantrips as the main source of damage, I will not allow me to use the PAM bonus attack. That is until level 7 when I use the cantrips until Level 7 where EK gets War Magic

Beginning at 7th level, when you use your action to cast a cantrip, you can make one weapon attack as a bonus action.

However, I will also be losing the Extra Attack feature since I will not be able to attack twice when using a cantrip. Then scaling past level 10, I will lose even more Extra Attacks.

My questions are: a) Is that how Extra Attack and War Magic interact? In that regard, is it worth becoming an Eldritch Knight?

b) Other than thematically, does it make sense in terms of damage to use 3 feats for what appears to be a less than synergistic character, between normal attacks, cantrip attacks and extra attacks?

c) Should the natural progression be level 1 PAM, level 4 SS and then level 6 WC?

Again, thank you for taking the time to read this and reply!

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    \$\begingroup\$ Does this answer your question? Does Booming Blade and Spell Sniper Stack? \$\endgroup\$ Dec 15 '19 at 14:53
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    \$\begingroup\$ Highly related: Can you combine Polearm Master with War Caster to cast Booming Blade as enemies approach? \$\endgroup\$
    – Someone_Evil
    Dec 15 '19 at 15:14
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    \$\begingroup\$ Also, Welcome to RPG.SE! Take the tour if you haven't already and see the help center or ask us here in the comments (use @ to ping someone) if you need more guidance. Good Luck and Happy Gaming! \$\endgroup\$
    – Someone_Evil
    Dec 15 '19 at 15:14
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    \$\begingroup\$ The question linked by @Revolver_Ocelot should cover whether this works by the rules. If you agree, please edit down the question to only be about the optimization aspect (whether it is worth it). If you are unsure about any of this, feel free to ask. \$\endgroup\$
    – Someone_Evil
    Dec 15 '19 at 15:25
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    \$\begingroup\$ Please make any one question post about one question. It makes the process of the site work so much better. So please split them up (and link between them if it helps with context). Also, please don't get around closures by reposting the question. If a question is edited to no longer be a duplicate, it can (and will) be reopened. \$\endgroup\$
    – Someone_Evil
    Dec 15 '19 at 20:03