help in dumbing down proof of basic principle of counting.

- nincompoop

help in dumbing down proof of basic principle of counting.

- katieb

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- danica518

whats the basic principle of counting

- danica518

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

- 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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## More answers

- 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

proof ...

- danica518

|dw:1437241044215:dw|

- danica518

seen in tree diagram

- danica518

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

- danica518

you imagine each outcome of N turning into M outcomes

- danica518

so ud have M, N times

- nincompoop

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

- 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

That listing of table will do for proof

- ganeshie8

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

- nincompoop

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

- danica518

|dw:1437241650123:dw|

- danica518

well to me the tree is pretty much a proof too

- nincompoop

I was just doing that last drawing you did

- danica518

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

- nincompoop

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

- danica518

dice is easiest

- 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

its like his world split into 6 new worlds,

- danica518

LOL

- 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

|dw:1437242326430:dw|

- danica518

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

- nincompoop

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

- anonymous

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

- nincompoop

CORRECT

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