Expansion of common powers

(a+b)^0=1
(a+b)^1=a+b
(a+b)^2=a^2+2ab+b^2
(a+b)^3=a^3+3a^2b+3ab^2+b^3
(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b
(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5
(a+b)^6=a^6+6a^5b+15a^4b^2+20a^3b^3+15a^2b^4+6ab^5+b^6
…

Pascal’s triangle

The coefficients in the expansions above correspond to the rows of Pascal’s triangle:

row |
  0 |              1
  1 |            1   1
  2 |          1   2   1
  3 |        1   3   3   1
  4 |      1   4   6   4   1
  5 |    1   5  10  10   5   1
  6 |  1   6  15  20  15   6   1
       ^                   ^
       |_ column 0         |_ column 5

We can write the the coefficient at row r and column c as ^rC_c.

Finding specific terms using a calculator

If we want to find the coefficient of the b^3 term of the expansion of (a+b)^6, we can solve:

^6C_3=20

Finding a specific term using factorials

If we want to find ^nC_r, we can also use this formula:

^nC_r = \frac{n!}{r!(n-r)!}

Finding all the coefficients

(a+b)^n = {^nC_0 a^n b^0} + {^nC_1 a^{n-1} b^1} + {^nC_2 a^{n-2} b^2} + ... + {^nC_n a^0 b^n}

Expand (2+x)^4

(2+x)^4 = {^4C_0 (2)^4 (x)^0} + {^4C_1 (2)^3 (x)^1} + {^4C_2 (2)^2 (x)^2} + {^4C_3 (2)^1 (x)^3} + {^4C_4 (2)^0 (x)^4}
= 1 \times 16 \times 1 + 4 \times 8 \times x + 6 \times 4 \times x^2 + 4 \times 2 \times x^3 + 1 \times 1 \times x^4
= 16 + 32x + 24x^2 + 8x^3 + x^4

Expand (1+2x)^6

=1^6
+^6C_1(1)^5(2x)^1
+ ^6C_2(1)^4(2x)^2
+^6C_3(1)^3(2x)^3
+ ^6C_4(1)^2(2x)^4
+ ^6C_5(1)^1(2x)^5
+ ^6C_6(1)^0(2x)^6
=1 + 6 \times 1 \times 2x + 15 \times 1 \times 4x^2 + 20 \times 1 \times 8x^3 + 15 \times 1 \times 16x^4 + 6 \times 1 \times 32x^5 + 1 \times 1 \times 64x^6
=1 + 12x + 60x^2 + 160x^3 + 240x^4 + 192x^5 + 64x^6
Answer: 1 + 12x + 60x^2 + 160x^3 + 240x^4 + 192x^5 + 64x^6

Representing the number of combinations

We can write the number of combinations of choosing r items from a list of n items as either ^nC_r or \begin{pmatrix}n\\r\end{pmatrix}.

Common combinations

\begin{pmatrix}n\\0\end{pmatrix}=\begin{pmatrix}n\\n\end{pmatrix}=1

\begin{pmatrix}n\\1\end{pmatrix}=\begin{pmatrix}n\\n-1\end{pmatrix}=n

Expansion of common powers