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whats the basic principle of counting
1+1=2 2+1=3 3+1=4
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.
if there are n possible outcomes and for each of the n outcomes theres another m that means there has to be n*m
seen in tree diagram
i think its closely related to understanding the basic ideas of multiplication
you imagine each outcome of N turning into M outcomes
so ud have M, N times
\( (1,1), (1,2) ... (1,n)\) \( (2,1), (2,2) ... (2,n)\) \( (m,1), (m,2) ... (m,n)\)
This amounts to proving the cardinality of cartesian product of two finite sets is same as the product of cardinalities of individual sets : \[n(A\times B) = n(A)\times n(B)\]
That listing of table will do for proof
Easy to see that there are m*n entries in that table
and the tree is suppose to help build intuition or some sort
well to me the tree is pretty much a proof too
I was just doing that last drawing you did
its saying the same thing, u have n branches and each branch splitting into m parts again
now, let us make it more concrete by providing some real-world example.
dice is easiest
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
its like his world split into 6 new worlds,
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
in one world the heads was the beginning of all things, and in the other the tail
I think that is when the tree gives a better illustration of what is going on.
eh, it usually follows immediately from the definition of multiplication of natural numbers