Answer:
A
Step-by-step explanation:
We want to expand: [tex](5 + z)^4[/tex]. We need to use the Binomial Theorem for this, which states that for a monic expression like x + k (where k is a constant) to the nth power, the coefficients of the terms will follow the terms in the nth row of the Pascal Triangle. These terms in the Pascal Triangle are of the form:
nC0 , nC1 , nC2 , .... , nC(n - 1) , nCn ---- here, nCk is combinations: [tex]\frac{n!}{k!(n-k)!}[/tex] , where ! denotes factorial
So, since the power here is 4, we will be using: 4C0, 4C1, 4C2, 4C3, and 4C4 as our coefficients of the terms.
Let's make 5 + z into z + 5. Now, we expand:
[tex](z+5)^4=4C0 * z^4+4C1*z^3*5+4C2*z^2*5^2+4C3*z*5^3+4C4*5^4[/tex]
4C0 = 1 , 4C1 = 4 , 4C2 = 6 , 4C3 = 4 , and 4C4 = 1:
[tex]1 * z^4+4*z^3*5+6*z^2*5^2+4*z*5^3+1*5^4=z^4+20z^3+150z^2+500z+625[/tex]
Thus, the answer is A.
Hope this helps!
