Lottery odds calculated in your head, or pen and paper. So I am working out the odds for a lottery, picking 4 numbers between 1-35.
The equation is:
$$\mbox{odds}=\frac{35\cdot 34\cdot 33\cdot 32}{1\cdot 2\cdot 3\cdot 4}=52360$$
Yes, I can work this out on a calculator with ease. 
However, how can I work this out on pen and paper, or in my head with ease?
Are there any type of methods or cheats I could use to calculate this quickly?
 A: If you're happy with something very approximate, that you can do in your head, $35\times 34\times 33\times 32$ is about $33^4$. $33^2$ is about $1,100$, so the numerator is about $1.2$ million.  The denominator is $1\times 2\times 3\times 4$ is about $25$, which is $100/4$ so the answer is about $1.2$ million divided by a hundred, times four, which is $48,000$. 
A: $\frac{35\cdot34\cdot33\cdot32}{2\cdot3\cdot4}=35\cdot17\cdot11\cdot8=595\cdot11\cdot8=6545\cdot8=52360$
A: On pen and paper, this is just a few multiplies.  It is good to cancel factors first, getting $35*34*11*4$  Then look for easy ones.  I would now go to $140*34*11$.  Since multiplying by $11$ is just an addition, I would next to $140*34$  
In your head, you probably won't get an exact answer, so look for approximations. I would say $35/4! \approx 1.5$ and multiplying two of the other factors gives about $1000$, so we have $1.5*34*1000 \approx 51000$  Knowing lots of arithmetic facts helps a lot.
A: I tend to use "completing the squares" a lot for exact mental calculations.
$35.34.44 = 35.(39-5).(39+5)=35.(39.39-5.5)$
$= 35.(40.40-40-39-25)=35.(1521-25)=35.(1500-4)$
$=7.5(1500-4)=7.(7500-20)$
$=49000+3500-140=52500-140=52360$
Yes, superficially this looks quite long. However:


*

*Each step is very tiny

*Very little working memory is used 

A: $${35\cdot34\cdot33\cdot32\over1\cdot2\cdot3\cdot4}={5\cdot7\cdot2\cdot17\cdot3\cdot11\cdot8\cdot4\over3\cdot8}=10\cdot7\cdot17\cdot11\cdot4$$
The factor of $10$ will just give an extra $0$ at the end.  The "difficult" multiplications are
$$7\cdot17=70+49=119$$
$$119\cdot11=1190+119=1309$$
and
$$1309\cdot4=5236$$
so the final answer is $52360$.  
The trickiest part (for me) was getting the carry correct when adding $1190+119$.  Alternatively, you can do the first two multiplications as
$$17\cdot11=170+17=187$$
and
$$7\cdot187=7(200-13)=1400-91=1309$$
A: Update: I got a very good Idea from comments so I updated the answer:
$$T=\large\require{cancel}\frac{35\cdot34\cdot\color{purple}{\cancel{33}^{11}}\cdot\color{blue}{\cancel{32}^{\color{red}{\cancel{8}^{4}}}}}{1\cdot\color{red}{\cancel{2}}\cdot\color{purple}{\cancel{3}}\cdot\color{blue}{\cancel{4}}}=35\cdot34\cdot44$$
Now a good fact is square of a two digit number can be done by multiplying the tens digit by its successor and just appending 25 at the end like $35^2=\widehat{3\cdot4}\;25=1225$
$$\large35\cdot34\cdot44=(35^2-35)\cdot44=(1225-35)\cdot44=1190\cdot44$$
So:
$$\large T=1190\cdot44\\\large=1190\cdot4\cdot11\\\large=(4000+400+360)\cdot11\\\large=4760\cdot11\\\large=(47600+4760)=52360$$
