nincompoop one year ago help in dumbing down proof of basic principle of counting.

1. danica518

whats the basic principle of counting

2. danica518

1+1=2 2+1=3 3+1=4

3. nincompoop

If one experiment can result in any of m possible outcomes and if another experiment can result in any n possible outcomes then there are $$n \times m \rightarrow nm$$ possible outcomes of the two experiment.

4. danica518

if there are n possible outcomes and for each of the n outcomes theres another m that means there has to be n*m

5. nincompoop

proof ...

6. danica518

|dw:1437241044215:dw|

7. danica518

seen in tree diagram

8. danica518

i think its closely related to understanding the basic ideas of multiplication

9. danica518

you imagine each outcome of N turning into M outcomes

10. danica518

so ud have M, N times

11. nincompoop

$$(1,1), (1,2) ... (1,n)$$ $$(2,1), (2,2) ... (2,n)$$ $$(m,1), (m,2) ... (m,n)$$

12. ganeshie8

This amounts to proving the cardinality of cartesian product of two finite sets is same as the product of cardinalities of individual sets : $n(A\times B) = n(A)\times n(B)$

13. ganeshie8

That listing of table will do for proof

14. ganeshie8

Easy to see that there are m*n entries in that table

15. nincompoop

and the tree is suppose to help build intuition or some sort

16. danica518

|dw:1437241650123:dw|

17. danica518

well to me the tree is pretty much a proof too

18. nincompoop

I was just doing that last drawing you did

19. danica518

its saying the same thing, u have n branches and each branch splitting into m parts again

20. nincompoop

now, let us make it more concrete by providing some real-world example.

21. danica518

dice is easiest

22. danica518

1,2,3,5,6 so lets say a kid considers okay all these cases occurred when he threw the dice 1 2 3 4 5 6 now he will be like if i throw the dice again, theres 6 new cases for each of these possiblities 1, 123456 2, 123456 3, 123456 . . 6, 123456

23. danica518

its like his world split into 6 new worlds,

24. danica518

LOL

25. Empty

Flipping a coin and rolling a die. There will be 12 possible outcomes. You can have 1-6 with heads and 1-6 with tails. :P

26. danica518

|dw:1437242326430:dw|

27. danica518

in one world the heads was the beginning of all things, and in the other the tail

28. nincompoop

I think that is when the tree gives a better illustration of what is going on.

29. anonymous

eh, it usually follows immediately from the definition of multiplication of natural numbers

30. nincompoop

CORRECT