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Algebra2D

Sequences and Series - Theory & Concepts - Binomial Theorem

Arithmetic and geometric progressions, infinite series, summation notation, and the binomial theorem.

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Binomial Theorem & Pascal's Triangle

Binomial Power

(a+b)4(a + b)^{4}
Power (n)n=4n = 4
Selected Term Coefficient index (k)k=2k = 2

Combinatorial Formulation

(42)=4!2!(42)!=6\binom{4}{2} = \frac{4!}{2!(4-2)!} = 6

This coefficient represents the term containing ankbka^{n - k}b^{k} in the expanded algebraic series.

Pascal's Triangle

1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
56
28
8
1

Expansion Series

(a+b)4=a4+4a3b+6a2b2+4ab3+b4(a+b)^{4} = a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}