Ashe is the single best AD carry in the game. Her presence in any team fight changes the dynamics dramatically as she has so much utility at her disposal via AOE slows, single target slows, and global stuns. If you want become a competitive player in League of Legends, learning Ashe is one of the best things you can do to improve your game.
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Team fighting is what separates mediocre Ashe players from great Ashe players. Positioning and battle awareness is key. Never put yourself in danger, Ashe has multiple tools at her disposal to fend off bruisers and tanks who try to run Ashe out of battle. Keep an eye out and know when you are safe to fire auto attacks and when to just run away to maximize distance from potential threats that arise from being CC'd. There is a mentality that plagues lesser players where everyone believes that you should be focusing the enemy carry. As Ashe, this should rarely happen since Ashe should be hitting anyone she can safely, which usually ends up being tanks, support, or AP champs. If you find yourself against a team with a large number of stuns and snares, it is almost always best to just attack whoever is in range(and then only if it is safe).
Team fighting with Ashe is different than team fighting with other AD carries. The team fight should be limited around Ashe as she must be considered a potential threat in the battle. If Ashe is driven away from the fight and isolated she is no use as she cannot kill tanks nearly as fast as other AD carries. Ashe should be auto attacking a target and your team should be supporting you close by. If someone has a gap closer and catches you off guard your team can back you up and you can still remain in the fight. If you find yourself in games where 4 of your partners rush the enemy carry and 4 of the enemy team rushes you, remind your team to protect you since you are the carry.
Ideally Ashe's ultimate should be used to initiate team fights. However, it is rare for your team to always be winning and on the offensive. For defensive situations you should always look to arrow someone who is somewhat in front of the team and vulnerable to burst damage. The enemy should also lack cleanse or merc treads so the stun will last longer. Let your team know you are about to arrow so they will react in time, not everyone thinks on the same page even at the top level of play. Remember to use judgement to know that your team isn't going to get aced because you initiated a horrible team fight.
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Ashe is the single best AD carry in the game, but also the most complex to play at the top level. There is a fine line between maximizing DPS in addition to staying alive during team fights as an auto attacking range carry. I went over some tips on how to play Ashe in game but obviously I could not go over everything. The more you play Ashe the more you will learn how to adapt to various situations. Use your head and improve from your mistakes and eventually you will hopefully make no more.