"У меня десять пальцев."
Translation:I have ten fingers.
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It's in the genitive plural case. The reason for this is the number that precedes it. Russian has complex rules about declining things that are being counted. When there is ten of something, you use the genitive plural where you would usually use the nominative. Please read this post that explains how to properly decline numerals: https://www.duolingo.com/comment/12185422
The л is soft in the nominative, because it is followed by an e. Apparently, this is one of the words where the e disappears in most cases (didn't find any rules for that yet, guess you'll just have to learn them). In the genitive plural, to keep the l soft now it is followed by a ц, you'll have to add a soft sign. -ев is the normal genitive plural ending for words ending in unstressed ц. Disclaimer: I'm not a native speaker.