Find the largest coefficient in this expansion of a binomial 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.
 A: 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.
A: More generally,
let's look at
$(x+c)^n$
where $c$ is a constant.
The expansion is
$(x+c)^n
=\sum_{k=0}^n \binom{n}{k}x^k c^{n-k}
$.
To find the largest term,
look at the ratio of consecutive terms:
Let
$t(n, k)
=\binom{n}{k} c^{n-k}
$.
$\begin{array}\\
r(n, k)
&=\frac{t(n, k+1)}{t(n, k)}\\
&=\frac{\binom{n}{k+1} c^{n-(k+1)}}{\binom{n}{k} c^{n-k}}\\
&=\frac{\frac{n!}{(k+1)!(n-(k+1))!}c^{-(k+1)}}{\frac{n!}{k!(n-k)!}c^{-k}}\\
&=\frac{\frac{1}{(k+1)}c^{-(k+1)}}{\frac{1}{(n-k)}c^{-k}}\\
&=\frac{n-k}{c(k+1)}\\
\end{array}
$
For example,
$r(n, 0)
=\frac{n}{c}
$,
$r(n, 1)
=\frac{n-1}{2c}
$,
$r(n, n-1)
=\frac{1}{nc}
$.
We see that
initially,
$r(n, k)
> 1$
and,
eventually,
$r(n, k) < 1$.
This means that
the $t(n, k)$
initially increase
and then decrease.
This change happens
when
$r(n, k)
\sim 1$;
i.e.,
when
$n-k
=c(k+1)
$
or
$n-c
=k(1+c)
$
or
$k
=\frac{n-c}{1+c}
=k_0(n, c)
$.
For 
$k <
k_0(n, c)
$,
the terms increase,
and for
$k >
k_0(n, c)
$,
the terms decrease.
(Note that,
if $n \le c$,
the terms never increase,
so the first term
is the largest.)
Therefore,
the maximum term
is within one
of
$k_0(n, c)
$.
As to the precise one,
and the possibility that
$r(n, k_0(n, c))
= 1
$,
those
I'll leave for others.
A: By Binomial Expansion,
$(x+y)^n$ = ${n \choose 0}x^n y^0 + {n \choose 1}x^{n-1}y^1 + {n \choose 2}x^{n-2}y^2 + \cdots + {n \choose n-1}x^1 y^{n-1} + {n \choose n}x^0 y^n$
The problem now boils down to finding the max value of $10\choose k$*$4^k$ as $k$ varies from $0$ to $10$.
