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What is the model and make of your tablet ?
Go to Settings > Apps or Applications or Applications Manager. Swipe to All tab (next to Downloaded, Running etc tabs). Find Google Services Framework. Select it and force stop it. Then tap Clear Data and Clear Cache.
Then find Play Store. Select it. Tap Clear Data. Restart your tablet.
Then check Play Store and try downloading some apps. Please let me know if that didn't work, so we can continue troubleshooting.