The borders of Asia and Europe aren't set in stone yet, so an exact count is hard. It's definitely 50 or more though, which includes Australia and the islands in the Pacific which is still said to be part of Asia.

About 60 countries and protectorates Abkhazia Afghanistan Akrotiri and Dhekelia Armenia Azerbaijan Bahrain Bangladesh Bhutan British Indian Ocean Territory Brunei Cambodia People's Republic of China Republic of China Christmas Island Islands Cyprus Egypt Georgia Hong Kong India Indonesia Iran Iraq Israel Japan Jordan Kazakhstan North Korea South Korea Kuwait Kyrgyzstan Laos Lebanon Macau Malaysia Maldives Mongolia Burma Myanmar Nagorno-Karabakh Nepal Oman Palestine Philippines Qatar Russia Saudi Arabia Singapore Sri Lanka South Ossetia Syria Tajikistan Thailand Timor-Leste Turkey Turkish Northern Cyprus Turkmenistan United Arab Emirates Uzbekistan Vietnam Yemen

these are the ones i found. Abkhazia Afghanistan Akrotiri and Dhekelia Armenia Azerbaijan Bahrain Bangladesh Bhutan British Indian Ocean Territory Brunei Cambodia People's Republic of China Republic of China Christmas Island Islands Cyprus Egypt Georgia Hong Kong India Indonesia Iran Iraq Israel Japan Jordan Kazakhstan North Korea South Korea Kuwait Kyrgyzstan Laos Lebanon Macau Malaysia Maldives Mongolia Burma Myanmar Nagorno-Karabakh Nepal Oman Palestine Philippines Qatar Russia Saudi Arabia Singapore Sri Lanka South Ossetia Syria Tajikistan Thailand Timor-Leste Turkey Turkish Northern Cyprus Turkmenistan United Arab Emirates Uzbekistan Vietnam Yemen

I think it's like 48 countries

In the political sense of the word "country", there are 43 sovereign states *in* Asia, 5 more that straddle Asia and another continent (though most people would leave out Egypt as it's Sinai toehold doesn't count for much), 4 de facto states, 22 officially autonomous regions/states, and 4 Special Administrative Regions. ASIAN SOVEREIGN STATES Afghanistan Armenia Azerbaijan Bahrain Bangladesh Bhutan Brunei Cambodia China Cyprus East Timor Georgia India Iran Iraq Israel Japan Jordan Kirgizstan Kuwait Laos Lebanon Malaysia Maldives Mongolia Myanmar N. Korea Nepal Pakistan Qatar S. Korea Saudi Arabia Singapore Sri Lanka Taiwan Tajikistan Thailand The Philippines Turkmenistan U.A.E. Uzbekistan Vietnam Yemen TRANSCONTINENTAL STATES Egypt (Asia & Africa) Indonesia (Asia & Oceania) Kazakhistan (Asia & Europe) Russian Federation (Asia & Europe) Turkey (Asia & Europe) AUTONOMOUS STATES Kurdistan (Iraq) Guangxi Zhuang (PRC) Inner Mongolia (PRC) Ningxia (PRC) Tibet (PRC) Xinjiang (PRC) Nakhichevan (Azerbaijan) Aceh (Indonesia) Azad Kashmir (Pakistan) Muslim Midnanao (Philippines) Jeju-do (S. Korea) Gorno-Badkakhshan (Tajikistan) Karakaipakstan (Uzbekistan) Sakha Republic (Russia) Buryatia Republic (Russia) Altai Republic (Russia) Tuva Republic (Russia) Khakasia Republic (Russia) Chukotka Okrug (Russia) Jewish Oblast (Russia) Yamalia Okrug (Russia) Khantia-Mansia Okrug (Russia) SPECIAL ADMINISTRATIVE REGIONS Hong Kong (PRC) Macau (PRC) Cordillera Administrative Region (Philippines) Sinuiju (N. Korea) (*No, I didn't just do this for this answer. I had it already on file: I teach Asian Geography to Thai 8th Graders.) In the ethnic sense ('country' = a homeland of an ethnic group) there are hundreds, and many of them divided/disputed, e.g., Kashmir, Punjab, Malayan Thailand, Turkish Kurdistan, and Turkish Armenia.