Binomial`s power There is a formula which links the power of

Binomial’s power There is a formula which links the power of binomials with binomial coefficients. The formula is: 𝑛 !!! !
𝑛 !!! !
𝑛 !!! !
𝑛
𝑛 ! !
𝑎+𝑏 ! =
𝑎 𝑏 +
𝑎 𝑏 +
𝑎 𝑏 + ⋯+
𝑎! 𝑏 !!! +
𝑎 𝑏 0
1
2
𝑛−1
𝑛
This formula can also be written: !
𝑎+𝑏
!
=
!!!
𝑛 !!! !
𝑎
𝑏 𝑘
and read as: “a plus b to the power of n equals the summation of n choose k times a to the power of n minus k times b to the power of k when k increments from 0 to n”. Let’s suppose that n=3: 3 !!! !
3 !!! !
3 !!! !
3 ! !
𝑎+𝑏 ! =
𝑎 𝑏 +
𝑎 𝑏 +
𝑎 𝑏 +
𝑎 𝑏
0
1
2
3
= 𝑎! + 3𝑎! 𝑏 + 3𝑎𝑏 ! + 𝑏 ! We can get immediately the results of the binomial coefficients through the 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 The second row of the triangle gives us the values of binomial coefficients when n=1, 1
1
that means and which are both equal to 1. 0
1
The third row of the triangle gives us the values of binomial coefficients when n=2, 2
2
2
that means , and which are equal to 1, 2 and 1. 0
1
2
The fourth row of the triangle gives us the values of binomial coefficients when n=3, 3
3 3
3
that means , ,
and which are equal to 1, 3, 3 and 1. 0
1 2
3