nincompoop
  • nincompoop
help in dumbing down proof of basic principle of counting.
Mathematics
katieb
  • katieb
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danica518
  • danica518
whats the basic principle of counting
danica518
  • danica518
1+1=2 2+1=3 3+1=4
nincompoop
  • 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.

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danica518
  • 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
nincompoop
  • nincompoop
proof ...
danica518
  • danica518
|dw:1437241044215:dw|
danica518
  • danica518
seen in tree diagram
danica518
  • danica518
i think its closely related to understanding the basic ideas of multiplication
danica518
  • danica518
you imagine each outcome of N turning into M outcomes
danica518
  • danica518
so ud have M, N times
nincompoop
  • nincompoop
\( (1,1), (1,2) ... (1,n)\) \( (2,1), (2,2) ... (2,n)\) \( (m,1), (m,2) ... (m,n)\)
ganeshie8
  • ganeshie8
This amounts to proving the cardinality of cartesian product of two finite sets is same as the product of cardinalities of individual sets : \[n(A\times B) = n(A)\times n(B)\]
ganeshie8
  • ganeshie8
That listing of table will do for proof
ganeshie8
  • ganeshie8
Easy to see that there are m*n entries in that table
nincompoop
  • nincompoop
and the tree is suppose to help build intuition or some sort
danica518
  • danica518
|dw:1437241650123:dw|
danica518
  • danica518
well to me the tree is pretty much a proof too
nincompoop
  • nincompoop
I was just doing that last drawing you did
danica518
  • danica518
its saying the same thing, u have n branches and each branch splitting into m parts again
nincompoop
  • nincompoop
now, let us make it more concrete by providing some real-world example.
danica518
  • danica518
dice is easiest
danica518
  • 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
danica518
  • danica518
its like his world split into 6 new worlds,
danica518
  • danica518
LOL
Empty
  • 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
danica518
  • danica518
|dw:1437242326430:dw|
danica518
  • danica518
in one world the heads was the beginning of all things, and in the other the tail
nincompoop
  • nincompoop
I think that is when the tree gives a better illustration of what is going on.
anonymous
  • anonymous
eh, it usually follows immediately from the definition of multiplication of natural numbers
nincompoop
  • nincompoop
CORRECT

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