Binomial Theorem
Class 11 Mathematics · Chapter 7
CBSE Class 11 community
View all
Physics, Chemistry, Maths + English
7~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Every subject, one purchase- Mathematics
- Physics
- + 3 more
Physics, Chemistry, Biology + English
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Every subject, one purchase- Physics
- Chemistry
- + 3 more
Accountancy, Business Studies, Economics + English
7~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Every subject, one purchase- Accountancy
- Business Studies
- + 3 more
Mathematics
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
TrendingRevise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Accountancy
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Accountancy
Hindi Elective
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- हिंदी
Hindi Core
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- हिंदी
Chemistry
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Chemistry
Political Science
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Political Science
Urdu Elective
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- اردو
- Urdu Elective
Hindustani Music (Vocal / Melodic)
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- हिंदुस्तानी संगीत
Home Science
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Home Science
English Core
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Hornbill
- Snapshots
Geography
3~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 3 books, one purchase- Geography
Sociology
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Sociology
Economics
2~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Economics
Physics
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
All 2 books, one purchase- Physics
Physics
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Physics
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Chemistry
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Chemistry
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Biology
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Accountancy
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
TrendingRevise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Accountancy
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Business Studies
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Economics
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Economics
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
History
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Geography
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Geography
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Geography
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Political Science
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Political Science
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Sociology
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Sociology
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Psychology
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
English Core · Hornbill
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
English Core · Snapshots
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
English Elective · Woven Words
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
हिंदी (आधार)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
हिंदी (आधार)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
हिंदी (ऐच्छिक)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
हिंदी (ऐच्छिक)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
संस्कृतम् (केन्द्रिकम्)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
संस्कृत (ऐच्छिक)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
اردو (کور)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
اردو (اختیاری)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Urdu Elective
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Computer Science
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Informatics Practices
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
TrendingRevise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Home Science
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Home Science
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Fine Art · Painting
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Biotechnology
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Knowledge Traditions
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
सृजनात्मक लेखन और अनुवाद
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Health and Physical Education
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
हिंदुस्तानी संगीत (गायन / वादन)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
हिंदुस्तानी संगीत (गायन / वादन)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
हिंदुस्तानी संगीत (तबला / पखावज)
~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative
Revise all of it in about an hour- Space weighted by unit marks
- Charts from the book's own data
- Traps corrected
Summary
The binomial theorem gives an efficient way to expand \((a+b)^n\) for a positive integer \(n\) without repeated multiplication. The expansion is \((a+b)^n={}^{n}C_0a^n+{}^{n}C_1a^{n-1}b+{}^{n}C_2a^{n-2}b^2+\cdots+{}^{n}C_n b^n\), written compactly as \(\sum_{k=0}^{n}{}^{n}C_k a^{n-k}b^k\). It has \(n+1\) terms; the power of \(a\) decreases from \(n\) to \(0\) while the power of \(b\) increases from \(0\) to \(n\), and the index sum stays \(n\) in every term. The coefficients \({}^{n}C_k\) are the binomial coefficients, the same numbers that form Pascal's triangle, where each entry is the sum of the two above it. Special cases include \((1+x)^n=\sum {}^{n}C_k x^k\), so that putting \(x=1\) gives \(\sum {}^{n}C_k=2^n\) and \(x=-1\) gives the alternating sum \(0\). The binomial coefficients are generated by the Pascal rule \({}^{n}C_r+{}^{n}C_{r-1}={}^{n+1}C_r\), so each row of the triangle can be written from the one above without expanding. The theorem is proved by the principle of mathematical induction. It is used to compute powers like \((98)^5\) and \((101)^6\), compare magnitudes such as \((1.01)^{1000000}\) versus \(10000\), and prove divisibility results, for instance that \(6^n-5n\) leaves remainder 1 when divided by 25.
Binomial theorem for positive integral indicesBinomial coefficientsPascal's triangleGeneral observations on the expansionSpecial cases \((1\pm x)^n\)Applications: evaluation and divisibility
Key terms
Binomial theorem
\((a+b)^n=\sum_{k=0}^{n}{}^{n}C_k a^{n-k}b^k\) for positive integer \(n\).
Binomial coefficient
The coefficient \({}^{n}C_k=\dfrac{n!}{k!\,(n-k)!}\) of each term.
Pascal's triangle
A triangular array of binomial coefficients; each entry is the sum of the two directly above it.
Number of terms
The expansion of \((a+b)^n\) has \(n+1\) terms.
Index sum
In each term the powers of \(a\) and \(b\) add up to \(n\).
Special case
\((1+x)^n={}^{n}C_0+{}^{n}C_1 x+\cdots+{}^{n}C_n x^n\).
Extra questions & answers
Try one from this chapter
The number of terms in the expansion of \((a+b)^n\) is: