IB Mathematics Analysis and Approaches Question Type 3: Working with Expressions of Factorials Exercises

For $x=3$, $y=2$, and $k=3$, evaluate: $$\frac{(x!y!)!}{(x!y!-k)!}$$

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

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

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

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

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

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

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

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