Evaluate the following factorial expressions: a) 5! b) 8!/5! c) (10-3)!/(9-6)! d) (n-1)!/(n-3)! e) (n+2)!/(n+2)

$$5! = 120$$

The value is $\boxed{120}$.

b) Evaluate $\frac{8!}{5!}$

$$8 \times 7 \times 6 = 336$$

The value is $\boxed{336}$.

c) Evaluate $\frac{(10-3)!}{(9-6)!}$

$$(10-3)! = 7!$$ $$(9-6)! = 3!$$

$$7 \times 6 \times 5 \times 4 = 840$$

The value is $\boxed{840}$.

d) Evaluate $\frac{(n-1)!}{(n-3)!}$

$$(n-1)(n-2)$$

The value is $\boxed{(n-1)(n-2)}$.

e) Evaluate $\frac{(n+2)!}{n+2}$

$$(n+1)!$$

The value is $\boxed{(n+1)!}$.

f) If $\frac{1}{7!} + \frac{1}{8!} = \frac{x}{9!}$, find the value of $x$.

$$\frac{1}{7!} = \frac{1 \times 8 \times 9}{7! \times 8 \times 9} = \frac{72}{9!}$$ $$\frac{1}{8!} = \frac{1 \times 9}{8! \times 9} = \frac{9}{9!}$$

$$\frac{72 + 9}{9!} = \frac{x}{9!}$$ $$\frac{81}{9!} = \frac{x}{9!}$$

$$x = 81$$

The value of $x$ is $\boxed{81}$.

Examples: Exponent of a Prime P in (n)!
The exponent of a prime $p$ in $n!$ is given by Legendre's formula:

$$E_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \dots$$

a) Find the exponent of 5 in $50!$

$$E_5(50!) = \left\lfloor \frac{50}{5} \right\rfloor + \left\lfloor \frac{50}{5^2} \right\rfloor + \left\lfloor \frac{50}{5^3} \right\rfloor + \dots$$ $$E_5(50!) = \left\lfloor \frac{50}{5} \right\rfloor + \left\lfloor \frac{50}{25} \right\rfloor + \left\lfloor \frac{50}{125} \right\rfloor + \dots$$

$$E_5(50!) = 12$$

The exponent of 5 in $50!$ is $\boxed{12}$.

b) Find the exponent of 3 in $10!$

$$E_3(10!) = \left\lfloor \frac{10}{3} \right\rfloor + \left\lfloor \frac{10}{3^2} \right\rfloor + \left\lfloor \frac{10}{3^3} \right\rfloor + \dots$$ $$E_3(10!) = \left\lfloor \frac{10}{3} \right\rfloor + \left\lfloor \frac{10}{9} \right\rfloor + \left\lfloor \frac{10}{27} \right\rfloor + \dots$$

$$E_3(10!) = 4$$

The exponent of 3 in $10!$ is $\boxed{4}$.

c) Find the exponent of 7 in $343!$

$$E_7(343!) = \left\lfloor \frac{343}{7} \right\rfloor + \left\lfloor \frac{343}{7^2} \right\rfloor + \left\lfloor \frac{343}{7^3} \right\rfloor + \left\lfloor \frac{343}{7^4} \right\rfloor + \dots$$ $$E_7(343!) = \left\lfloor \frac{343}{7} \right\rfloor + \left\lfloor \frac{343}{49} \right\rfloor + \left\lfloor \frac{343}{343} \right\rfloor + \left\lfloor \frac{343}{2401} \right\rfloor + \dots$$

$$E_7(343!) = 57$$

The exponent of 7 in $343!$ is $\boxed{57}$.

d) What is the highest power of 7 that can divide $5000!$ without leaving a remainder?
This is equivalent to finding the exponent of 7 in $5000!$.

$$E_7(5000!) = \left\lfloor \frac{5000}{7} \right\rfloor + \left\lfloor \frac{5000}{7^2} \right\rfloor + \left\lfloor \frac{5000}{7^3} \right\rfloor + \left\lfloor \frac{5000}{7^4} \right\rfloor + \left\lfloor \frac{5000}{7^5} \right\rfloor + \dots$$ $$E_7(5000!) = \left\lfloor \frac{5000}{7} \right\rfloor + \left\lfloor \frac{5000}{49} \right\rfloor + \left\lfloor \frac{5000}{343} \right\rfloor + \left\lfloor \frac{5000}{2401} \right\rfloor + \left\lfloor \frac{5000}{16807} \right\rfloor + \dots$$

$$E_7(5000!) = 832$$

The highest power of 7 that can divide $5000!$ is $7^{832}$.
The highest power is $\boxed{832}$.