Consider a polynomial in $x$ and $y$ $$P = ( 3 x + 4 y + 1 ) ^ { 5 } .$$ Let us denote the coefficient of $x ^ { n } y$ in the expansion of $P$ by $a _ { n }$, where $n$ is an integer. Note that $x ^ { 0 } = y ^ { 0 } = 1$. (1) Let us find the value of the coefficient $a _ { 1 }$. First, we note that $$P = \{ ( 3 x + 1 ) + 4 y \} ^ { 5 }$$ and use the binomial theorem to expand $P$. Then $x y$ appears when we expand the term AB $( 3 x + 1 ) ^ { \text {C} } y$. Further, the coefficient for $x$ in the expansion of $( 3 x + 1 ) ^ { \text {C} }$ is DE. It follows that $$a _ { 1 } = \mathbf { F G H } .$$ (2) The number of values which $n$ can take is $\square$ in all. Also, the value of $a _ { n }$ is maximized at $n =$ $\square$ J .
Consider a polynomial in $x$ and $y$
$$P = ( 3 x + 4 y + 1 ) ^ { 5 } .$$
Let us denote the coefficient of $x ^ { n } y$ in the expansion of $P$ by $a _ { n }$, where $n$ is an integer. Note that $x ^ { 0 } = y ^ { 0 } = 1$.
(1) Let us find the value of the coefficient $a _ { 1 }$. First, we note that
$$P = \{ ( 3 x + 1 ) + 4 y \} ^ { 5 }$$
and use the binomial theorem to expand $P$. Then $x y$ appears when we expand the term AB $( 3 x + 1 ) ^ { \text {C} } y$. Further, the coefficient for $x$ in the expansion of $( 3 x + 1 ) ^ { \text {C} }$ is DE. It follows that
$$a _ { 1 } = \mathbf { F G H } .$$
(2) The number of values which $n$ can take is $\square$ in all. Also, the value of $a _ { n }$ is maximized at $n =$ $\square$ J .