Arithmetic with Factorials

Question

The problem is the following (without using a calculator, reduce):

$$ {52! - 51!}\over{51! - 50!} $$

This is a simple problem, but I have not been able to find a good resource online explaining the intuition behind solving this.

My initial thoughts were to use something like:

$$ {52 \times 51 \times 50 \times 49 ... } \over {51 \times 50 \times 49 \times 48 ... } $$

and then cross out the common multiples to be left with something. It does not seem like simple arithmetic subtraction would get the right answer.

How do I go about solving this? Any advice would be greatly appreciated.

Edit:

Would some answer in the format of:

$ {(52)(52)} \over {49!} $

be on the right track?

Thanks.

Answer 1

$$ \frac{52! - 51!}{51! - 50!} = \frac{51!(52-1)}{50!(51-1)} = \frac{51(52-1)}{51-1} = \cdots $$

Answer 2

Hint:

Factor out the least factorial both in the numerator and the denominator: $$52!-51!=51!\,(52-1), \qquad 51!-50!= 50!\, (51-1).$$ Can you proceed?

Answer 3

Divide top and bottom by $50!$ to get:

$\dfrac{52\cdot51-51}{51-1}=\dfrac{2652-51}{50}=\dfrac{2601}{50}=52.02$

Answer 4

${51^2\cdot 50!\over 50 \cdot 50!}= {51^2\over 50} = {1\over 50}+52$

I'm not going to tell you how I got here, you'll have that part to figure out.