Question

How do you use the binomial series to expand `(1+ax)^10`?

Answer 1

`C(10,0) + C(10,1)a + C(10,2)a^2 + C(10,3)a^3 + C(10,4)a^4 + C(10,5)a^5 + C(10,6)a^6 + C(10,7)a^7 + C(10,8)a^8 + C(10,9)a^9 + C(10,10)a^10`, where C(10,i) is the binomial coefficient representing the number of different ways to choose i items from a set of 10, namely `(10!) / (i! (n-i)!)`.

Explanation:

In general, for `(x+y)^k`, the expansion is `C(k,0)x^ky^0 + C(k,1)x^(k-1)y^1 + C(k,2)x^(k-2)y^2 + .... C(k,k-1)x^1y^(k-1) + C(k,k)x^0y^k`.

Note that, for example, `C(4,0)=C(4,4)=1, C(4,1)=C(4,3)=4, C(4,2)=6`
and Pascal's triangle gives a diagram representing binomial coefficients, namely:

                                               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

etc.
The sequence of numbers in the i-th row are, respectively, `C(1,0), C(i,1), C(i,2), C(i,3),` etc.

The number `C(i,j)` is the sum of the two numbers `C(i-1,j-1)` and `C(i-1,j)` just above it in the diagram and to the left and right, respectively.