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This is an audio version of the Wikipedia Article:\nhttps://en.wikipedia.org/wiki/Binomial_theorem\n\n\n00:00:24 1 History
00:02:28 2 Theorem statement
00:05:56 3 Examples
00:12:12 3.1 Geometric explanation
00:21:55 4 Binomial coefficients
00:22:09 4.1 Formulae
00:27:09 4.2 Combinatorial interpretation
00:27:39 5 Proofs
00:30:05 5.1 Combinatorial proof
00:31:44 5.1.1 Example
00:31:53 5.1.2 General case
00:32:02 5.2 Inductive proof
00:34:57 6 Generalizations
00:36:26 6.1 Newton's generalized binomial theorem
00:36:39 6.2 Further generalizations
00:39:33 6.3 Multinomial theorem
00:41:42 6.4 Multi-binomial theorem
00:41:51 6.5 General Leibniz rule
00:50:33 7 Applications
00:55:59 7.1 Multiple-angle identities
00:59:48 7.2 Series for e
01:03:14 7.3 Probability
01:04:22 8 The binomial theorem in abstract algebra
01:04:44 9 In popular culture
01:04:53 10 See also
01:11:35 11 Notes
01:16:22 12 References
01:17:26 13 Further reading
01:18:06 14 External links
01:18:58 See also
\n\n\nListening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.\n\nLearning by listening is a great way to:\n- increases imagination and understanding\n- improves your listening skills\n- improves your own spoken accent\n- learn while on the move\n- reduce eye strain\n\nNow learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.\n\nListen on Google Assistant through Extra Audio:\nhttps://assistant.google.com/services/invoke/uid/0000001a130b3f91\nOther Wikipedia audio articles at:\nhttps://www.youtube.com/results?search_query=wikipedia+tts\nUpload your own Wikipedia articles through:\nhttps://github.com/nodef/wikipedia-tts\nSpeaking Rate: 0.856828981820191\nVoice name: en-US-Wavenet-D\n\n\n"I cannot teach anybody anything, I can only make them think."\n- Socrates\n\n\nSUMMARY\n=======\nIn elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form a xb yc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example (for n = 4),




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4


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x

4


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4

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3


y
+
6

x

2



y

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+
4
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4


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{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}
The coefficient a in the term of a xb yc is known as the binomial coefficient







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
or







(


n
c


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{\displaystyle {\tbinom {n}{c}}}
(the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
gives the number of different combinations of b elements that can be chosen from an n-element set.