[Math] Find the largest coefficient in this expansion of a binomial

Question

What is the largest coefficient in the expansion of $(x+4)^{10}$?

This is not a homework question, it's a question from a mathleague competition that I did not understand. Please keep answers at the high school mathematics level.

Answer 1

Binomial theorem states that the $k^{th}$ term (if we start counting from $0$) of the expansion is $$\binom{10}{k}\cdot 4^k \cdot x^{10-k}$$

The coefficient of any term is therefore going to be $\binom{10}{k}\cdot 4^k$. We will try to maximize this value. I will do this by calculator:

$k=10, 4^{10}= 1048576, \binom{10}{10}=1 \to 1048576$

$k=9, 4^{9}= 262144, \binom{10}{9}=10 \to 2621440$

$k=8, 4^{8}= 65536, \binom{10}{8}=45 \to 2949120$

$k=7, 4^{7}= 16384, \binom{10}{7}=120 \to 1966080$

$k=6, 4^{6}= 4096, \binom{10}{6}=210 \to 860160$

$k=5, 4^{5}= 1024, \binom{10}{5}=252 \to 258048$

After this, both the binomial coefficients and the power of $4$ will only decrease. Therefore, the largest coefficient is at $k=8$, and is 2949120.