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In the Hunger Games: Mockingjay - Part 2 when President Snow's execution comes and Katniss is given the honour to execute him with a special arrow and everyone is watching the execution, Katniss does not shoot Snow, in fact, she kills Coin.

Why did she do this if Coin was the President of District 13 and was on Coin's side and not kill Snow?

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Because it turns out that Coin is just as corrupt as President Snow. According to Wikipedia:

Katniss learns the Capitol has been conquered and Snow captured. When Katniss confronts Snow in his rose garden, he explains that Coin orchestrated the bombing using a Capitol hovercraft to make it look like he would kill their children and to turn his soldiers against him. … She realizes that the incident resembles a trap that Gale had developed earlier, bombing an area, then again later to kill medics. … At the execution, Katniss shoots and kills Coin instead of Snow, realizing that she is no better than Snow for killing people for her own personal gain.

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Katniss, ever the shrewd woman, realized that Coin and Snow are two sides of the same coin. That Coin was just as willing to continue the same dictatorship as Snow, keep the Hunger Games going just with Capital children, etc. If Coin took over, nothing would change. In fact, she learned that Coin ordered the attack on the Capital children that ended up killing her little sister Prim. And Coin had previously manipulated Katniss in District 13, was willing to execute Peeta. Coin was a villain masquerading as a righteous rebel leader.

So she killed the healthy tyrant, knowing that Snow was dead either way, from poison or from the crowd.

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