Ananya's First Question!

$\begin{aligned}\frac{99!\ \color{#D61F06}{8!} \ 6! \ 5!\ \color{#3D99F6}{4!}}{\color{#20A900}{100!}\ 7!\ 3!\ \color{#624F41}{0!}}=&\ \frac{99!\cdot\color{#D61F06}{8\cdot7!}\cdot6!\cdot5!\cdot\color{#3D99F6}{4\cdot3!}}{\color{#20A900}{100\cdot99!}\cdot7!\cdot3!\cdot\color{#624F41}1}\\=&\ \frac{8\cdot\color{teal}{6!}\cdot\color{magenta}{5!}}{\color{#302B94}{25}}\\=&\ \frac{8\cdot\color{teal}{720}\cdot\color{magenta}{120}}{\color{#302B94}{5\cdot5}}\\=&\ 8\cdot144\cdot24=\boxed{27648}\end{aligned}$

$\frac { 99!8!6!5!4! }{ 100!7!3!0! } =\frac { 99!8*7!6!5!4*3! }{ 100*99!7!3!0! } \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \frac { 8*6!5!*4 }{ 100 } \quad \quad \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad 27648\quad$

99!/100!= 1/100: 8!/7! = 8: 4!/3! = 4 Then (8x6!x5!x4)/100 = 27648