Answer
$ 300a^{2}b^{23}$
Work Step by Step
$(a+b)^{n}=\left(\begin{array}{l}
n\\
0
\end{array}\right)a^{n}+\left(\begin{array}{l}
n\\
1
\end{array}\right)a^{n-1}b+\left(\begin{array}{l}
n\\
2
\end{array}\right)a^{n-2}b^{2}+\cdots+\left(\begin{array}{l}
n\\
n
\end{array}\right)b^{n}$
$\displaystyle \left(\begin{array}{l}
n\\
r
\end{array}\right)=\frac{n!}{r!(n-r)!},$
$n!=n(n-1)\cdot...\cdot 2\cdot 1,\ \ $
$n!=0$
Properties:
$\left(\begin{array}{l}
n\\
r
\end{array}\right)=\left(\begin{array}{l}
n\\
n-r
\end{array}\right), $
$ \left(\begin{array}{l}
k\\
r-1
\end{array}\right)+\left(\begin{array}{l}
k\\
r
\end{array}\right)=\left(\begin{array}{l}
k+1\\
r
\end{array}\right)$
--------
$...$In all the terms, exponents of a and b add to n...
In $(a+b)^{25} $, which has 26 terms,
the last three terms contain powers of a:
24th, 25th, 26th:
$a^{2},\ \ \ a^{1},\ \ \ a^{0}$
The 24th term in the expansion of
$(a+b)^{25}$ is where
the exponent of $a$ is $2,$
the exponent of $b$ is $25-2=23,$ and
the coefficient is
$\left(\begin{array}{l}
25\\
23
\end{array}\right)= \left(\begin{array}{l}
25\\
25-23
\end{array}\right)= \displaystyle \left(\begin{array}{l}
25\\
2
\end{array}\right)=\frac{25\cdot 24}{1\cdot 2}=300$
The term is: $ 300a^{2}b^{23}$