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You can obtain permit at police check post or at Sikkim Tourism department for both the above spots. You need to submit two passport size photos along with passport or driving licence. Online facility are available but it needs to be stamped at Rangpo check post. For zero point you could obtain the permit at Chopta valley or Thangu check post. Apply separate permits and under Protected Area Permit.
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Do not have to worry, as you can easily get permit from the check posts near the mentioned location. you have to carry your ID proof- Passport/Driveing License/Voter card.
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