Question

What is the Binomial Expansion of `(d+3)^7`?

Answer 1

Recall the general formula of Binomial Expansion:

`(x+y)^N = C_N^0 x^N y^0 + C_N^1 x^(N-1) y^1 + C_N^2 x^(N-2) y^2 +...+ C_N^(N-2) x^2 y^(N-2) + C_N^(N-1) x^1 y^(N-1) + C_N^N x^0 y^N`

or

`(x+y)^N = Sigma_(k=0)^N C_N^k x^(N-k) y^k`

where `C_N^k=(n!)/(k!*(n-k)!)` - the number of combinations from `N` objects by `k`.

Using this formula for `x=d, y=3, N=7`, we obtain:
`(d+3)^7 = C_7^0 d^7 3^0 + C_7^1 d^6 3^1 + C_7^2 d^5 3^2 + C_7^3 d^4 3^3 + C_7^4 d^3 3^4 + C_7^5 d^2 3^5 + C_7^6 d^1 3^6 + C_7^7 d^0 3^7`

Calculating the binomial coefficient:
`(d+3)^7 = d^7 + 21d^6 + 189d^5 + 945d^4 + 2835d^3 + 5103d^2 + 5103d^1 + 2187`

If you want more information about Binomial Expansion, you can listen a lecture on Unizor by following the menu topics Math Concepts - Induction - Binomial where the a proof of the above binomial expansion formula is presented.