The Tu-154 plane crashed in the Black Sea shortly after taking off from the southern city of Adler where it had been refuelling, defence ministry spokesman Igor Konashenkov told Russian news agencies.
It disappeared from radars just two minutes after it took off at 5:25 am (0755 IST).
The ministry told agencies there was no sign of any survivors at the crash site and that four bodies had been recovered off the coast of the resort city of Sochi, after authorities launched a frantic search operation.
President Vladimir Putin has ordered Prime Minister Dmitry Medvedev to head a government commission to investigate the crash.
The plane had been on a routine flight to Russia's Hmeimim airbase in western Syria, which has been used to launch air strikes in Moscow's military campaign supporting its ally President Bashar al-Assad in the country's devastating civil war.
Among the plane's 84 passengers were Russian servicemen as well as 64 members of the Alexandrov Ensemble, the army's official musical group also known as the Red Army Choir, and its conductor Valery Khalilov. They were headed to Syria to participate in New Year celebrations at the airbase.
There were also eight crew members onboard, the ministry said. A list of passengers published by the defence ministry also included Elizaveta Glinka, a doctor and charity worker who serves on the Kremlin human rights council.
Mikhail Fedotov, who heads the council, said Glinka was travelling to Syria to bring medication to a university hospital in the coastal city of Latakia near the airbase, agencies reported.
Kremlin spokesman Dmitry Peskov told news agencies that Putin was being kept updated on the search operation and was in constant contact with defence minister Sergei Shoigu.
The ministry has not put forward any possible causes of the crash.
Konashenkov said that the aircraft had been in service since 1983 and had flown some 7,000 hours since. The plane last underwent repairs in December 2014 and was serviced in September, he said.
Russia's Investigative Committee said a criminal probe had been launched to determine whether violations of air transportation safety had led to the crash.
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