Can I do Android Programming in C++, C?

Yes, you can program Android apps in C++ (for the most part), using the Native Development Kit (NDK), although Java is the primary/preferred language for programming Android, and your C++ code will likely have to interface with Java components, and you'll likely need to read and understand the documentation for Java components, as well. Therefore, I'd advise you to use Java unless you have some existing C++ code base that you need to port and that isn't practical to rewrite in Java.

Java is very similar to C++, I don't think you will have any problems picking it up... going from C++ to Java is incredibly easy; going from Java to C++ is a little more difficult, though not terrible. Java for C++ Programmers does a pretty good job at explaining the differences. Writing your Android code in Java will be more idiomatic and will also make the development process easier for you (as the tooling for the Java Android SDK is significantly better than the corresponding NDK tooling)

In terms of setup, Google provides the Android Studio IDE for both Java and C++ Android development (with Gradle as the build system), but you are free to use whatever IDE or build system you want so long as, under the hood, you are using the Android SDK / NDK to produce the final outputs.

You should use Android NDK to develop performance-critical portions of your apps in native code. See Android NDK.

Anyway i don't think it is the right way to develop an entire application.

PLEASE NOTE: THE ANSWER BELOW IS HORRIBLY OUTDATED, AND MIGHT NOT BE ENTIRELY CORRECT ANYMORE.

You can program in C/C++ using the Android NDK. You'll have to wrap your c++ codebase in a static library and load that through a Java wrapper & JNI.

The standard NDK does not support RTTI and a lot of the functionality of standard c++ is also not available such as std::string, etc. To solve this you can recompile the NDK. Dmitry Moskalchuk supplies a modified version of the NDK that supports this at http://www.crystax.net/android/ndk-r3.php. This modified version works on all Android phones that run on an ARM processor.

Depending on the kind of application you should decide to use Java or C/C++. I'd use C/C++ for anything that requires above average computational power and games -- Java for the rest.

Just pick one language and write the majority of your application in that language; JNI calls can decrease your performance by a lot. Java isn't that hard though -- if you know how to program in C/C++. The advantage of Java is that the application can run on any device running Android, where as NDK applications depend on the architecture it was compiled for.