Question Type 3: Working with expressions of factorials Exercises

For $x=3$, $y=2$, and $k=3$, evaluate:

Simplify the factorial expression $\frac{(x!\,y!)!}{\bigl(x!\,y! - 1\bigr)!}\,.$

Show that $\frac{(x!\,y!+1)!}{\bigl(x!\,y! - 1\bigr)!}=(x!\,y!)\,(x!\,y!+1)\,.$

Express $\frac{(x!\,y!)!}{\bigl(x!\,y! - k\bigr)!}$ as a product of $k$ consecutive factors. Assume $1\le k\le x!\,y!$.

Simplify the expression $\frac{(x!\,y!)!}{\bigl(x!\,y! - 1\bigr)!\,(x-1)!\,(y-1)!}\,.$

Evaluate $\frac{(x!\,y!)!}{\bigl(x!\,y! - 1\bigr)!\,(x-1)!\,(y-1)!}$ for $x=2$, $y=3$.

Simplify the expression $\frac{(x!\,y!+2)!}{(x!\,y!)!}\,.$

Evaluate $\frac{(x!\,y!)!}{\bigl(x!\,y! - 1\bigr)!\,(x-1)!\,(y-1)!}$ for $x=3$, $y=4$.

Simplify the expression $\frac{(n+3)!}{n!}\,.$