a+b+c+d = k
a

Mathematics
**
**- dan815

a+b+c+d = k
a**
**

Mathematics
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- dan815

a+b+c+d = k
a**
**

Mathematics
- Stacey Warren - Expert brainly.com

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- dan815

@TrojanPoem

- TrojanPoem

Solve this in front of me. And give me another example

- dan815

okay umm lemme show u why i think this relation is important one

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- dan815

okay ill show u all my work so far, its not much lol

- TrojanPoem

:)

- dan815

okay so i started working backwards, now i allowed 0 to be in the first place of the digit, we can take care of the problems of that later on

- dan815

im looking at only the increasing case, and the decreasing case will be the same numbers

- dan815

so for 10 digit numbers we have 1 non bouncy number that is strictly increasing
0123456789
and 1 non bouncy number that is strickly decreasing
9876543210

- dan815

if there were 2 strickly increasing then there have to be 2 strictly decreasing so you can see why we only care about one half

- TrojanPoem

123 increasing
321 decreasing
456 inc
654 dec ( non bouncy) all )

- dan815

yes that is correct

- dan815

okay now lets move to the 9 digit case

- dan815

here is where we can see the pattern

- TrojanPoem

123456789
987654321

- dan815

leets only look at the increasing for now

- TrojanPoem

Ok.

- dan815

dont think about the decreasing its just a mirror and repeated

- TrojanPoem

Yeah, you just turn left to right and so on.

- dan815

okay umm wait this is important first

- dan815

if u look at the 10 digit case

- dan815

|dw:1435690693214:dw|

- dan815

there is 1 difference between all the digits,

- dan815

the sum of the difference have to add upto 9, and all of them have to be more than 0

- dan815

so u can see there is no way to arrange the 1s to get anything new that tells us there is infact only 1

- dan815

now if u look at the 9 digit, case
we have 8 difference now there is a max number we can pick for the first place

- TrojanPoem

What do you mean by all of them have to be more than 0 ? why not 0 ?

- dan815

which is 1, not 0 anymore, but anything more than 1 means we will fail

- dan815

because if the difference is 0 then we have 1 1 1 1 1 which is bouncy

- dan815

12345 all have diff 1 for their digits

- dan815

following so far?

- TrojanPoem

You mean the difference between the numbers can't be zero. Fine I understood all.

- dan815

okay now look at this

- TrojanPoem

The sum of difference 12345 is 4

- dan815

|dw:1435690935221:dw|

- dan815

so for the 9 digit case we know that 1 is the biggest number

- dan815

now if you pick 1, then all the following differences have to be 1, so we know starting with 1
there is only 1 number
123456789

- dan815

when you pick 1, the difference have to add up to 8 and we have 8 places so everything is 1

- dan815

but when you pick 0, the differences can add up to 9 and we have 8 places

- TrojanPoem

But if we used 2, won't the number above 23456789---
will be bouncy if we added any other digit ?

- dan815

yes that is why 2 is not possible

- dan815

are u good upto here?

- dan815

this is where the parition question starts

- TrojanPoem

Yeah. But you must give me 1 or 2 easy questions as not to lose the info I gained.

- dan815

okay i think we should just work on this simpler problem

- dan815

a+b+c+d = k a**
**

- dan815

lets say K = 7

- dan815

a+b+c+d=7
how many solutions are there to this?
where
a**
**

- TrojanPoem

Bouncy or non bouncy numbers ?

- dan815

no this is just a separate question

- dan815

you have to find hte number of integer solutions to that equation

- TrojanPoem

Lol a = 0 , b = 1, c= 2 , d= 4 or a = 0 , b = 0 , c = 0 , d = 7 there is infinity. Clarify your aimbition

- dan815

a**
**

- TrojanPoem

Oh

- TrojanPoem

a = 4 ,b = 2, c = 1 ,d = 0

- dan815

you have to find the number* of integer solutions

- TrojanPoem

a+ b = 2 , a < b < solve this.

- dan815

there is only 1 solution to that

- dan815

oh and a b c d > 0 ofcourse

- TrojanPoem

Ok .. because only when a = 2
But what about this a + b + c = 50 a < b

- dan815

or equal to 0

- dan815

yeah thats the question im asking u lol

- dan815

50 is a big one try just 7 for now

- TrojanPoem

Ok a = 4, b = 2, c = 1, d = 0 (1)
a = 5, b= 2, c =1 , d= =0
Maybe 2 or so

- dan815

i think a logical way to break this question down is to look at the differences

- dan815

a+b+c=7
okay
say a=0
this means we have a difference of 7 to distribute

- TrojanPoem

Ok , so failed because it doesn't satisfy the condition.

- dan815

0 , 1, 6
0,2,5
0,3,4
no other solutions with a = 0

- dan815

hmm okay u think about it for a bit, i am gonna look at something

- TrojanPoem

Damn -_- I saw the condition a > b > c >d

- dan815

thats okay too same thing

- TrojanPoem

But In your solution you made a = 0 and other number = 0 this is not possible you have broken the condition right ?

- dan815

xD dont get caught on details lol

- dan815

we need to work faster rn

- TrojanPoem

Lol, details is important. But what about a+b+c+d = 10^9
won't I die before completing it ?

- TrojanPoem

are*

- dan815

i found 11 increasing non bouncy numbers for digit 9 , im gonna work on 8 now

- dan815

try this problem instead ditch the older one
a + b+ c = 7
and a,b,c >0 and

- dan815

so we are allowing the equal case now

- TrojanPoem

Ok first a =0
b + c = 7
b = 0 c = 7
b =2 c = 5
b = 1 c = 6
b = 3 c = 4
b = 4 c = 3
b = 5 c = 2
b = 6 c = 1
Ok I got it , but still there is a lot a lot a lot of solutions

- TrojanPoem

I got a scheme on how to do it
first a = 0
b +c = 7 ( b from 0 to 7) and calculate c
then a = 1
b+c = 6 (b from 0 to 6) and calculate c
... until the end.

- dan815

yes thats where i am right now

- dan815

now instead of this, people have found a formula to do that counting for the number of solutions

- dan815

which is our goal

- TrojanPoem

AND YOU LEFT ME CALCULATING !!!!!!!!!!!!!!!!! >:(

- TrojanPoem

But at least I learnt the basics :P

- dan815

this stuff is an intro to something called paritioning theorems in mathematics

- dan815

people have been working on approximations formulas for this count

- TrojanPoem

approximations ? So answers are not accurate ?

- dan815

the degree of error is quite low, so for the number we are dealing with the error will not effect it

- TrojanPoem

What's that formula ?

- dan815

in other words its like people are finding formulas that are able to solve higher and higher values for k
a+b+c=k

- dan815

i dont know, there are a lot apparently

- dan815

okay but theres still work to be done here before the formula

- dan815

there are more simpler ways to count

- TrojanPoem

Like ?

- dan815

okay like take this case for 9 digit

- dan815

the difference if you pick 0 as the start

- dan815

are
1,1,1,1,1....,2
there is one of 2
that means there are 9 ways with a dfference of 2 and all 1s

- dan815

the other 2 cases are when all the differences are 1
which gives us 2 more cases so a total of 9

- dan815

now for the 8 digit case, we can use some combinatorics to count faster

- dan815

for a starting value of 1
we have 8 cases
for a starting value of 0
we can have a 3 somewhere or a 2 somewhere
so 16 more ways from that

- dan815

and so on
for 7

- TrojanPoem

explain " there is one 2"
we can have a 3 somewhere or a 2 somewhere

- dan815

if u look at the differences

- dan815

when u have 0 as a start

- dan815

|dw:1435693429832:dw|

- dan815

and we can have 2 and 2

- dan815

how many ways to distribute 2, 2s in 8 positions

- dan815

it would be 8*7/2

- TrojanPoem

I am dumb lol. |dw:1435693617081:dw|

- dan815

oops lol it should be 1 1, 2

- dan815

|dw:1435693673194:dw|

- TrojanPoem

And you didn't tell me where the 3 gone ?

- dan815

what do u mean?

- TrojanPoem

1 2 4 56789 ?

- dan815

no look at the differences

- dan815

we are trying to find the unique differences and see the number of permutatiosn each have

- dan815

there is 1 way to arrange all 1s
there are 8 ways to arrange one 2 and rest 1
there are 8 ways to arrange one 3 and rest 1
there are 8*7/2 ways to arrange two 2s and rest 1

- dan815

and that completes the case for a=0 for the 8 digit case

- dan815

for a=1 there are
8+1 way
for
a=2 there is 1 way
so a total of
1+8+8+8*7/2+9+1= eight digit CASE

- dan815

so everything is basic computation including the combinatorics we can make computer do, the one part we need to figure out is the unique solutions

- TrojanPoem

Give me simple exercise

- TrojanPoem

a simple*

- dan815

a+b+c=5
a,b,c>0

- dan815

find a nice solution to that and we will solve this whole quetion in like 0.0000000000001 computation time hehe

- dan815

a+b+c=5
a,b,c>0
Total number of solutions = ?

- TrojanPoem

hahahahahahahahahaaaaaaaaaaaa, Nice solution ? I am using dumbs way.

- dan815

here is something that might be useful im not sure yet

- dan815

if you take the 2nd differences we can allow 0 so this is great

- dan815

then we can use a nice combinatorics trick

- dan815

like look at this other question okay where we allow 0

- dan815

a+b+c = 10 lets say

- dan815

you can think of this problem like this think of the 10 as being 10 ones

- TrojanPoem

|dw:1435694528337:dw|

- dan815

1 1 1 1 1 1 1 1 1 1
and the number of places we can put 2 plus signs to get unique solution

- dan815

will tell us total number of solutions

- dan815

1 1 1 1+ 1 1 1 1+ 1 1
this is
saying
4 + 4 + 2

- TrojanPoem

Wait wait, here is my problem "put 2 signs to get unique solution"

- dan815

there are a total of 12 spots if u count the 1s and + sign as spots
so that means
12 choose 2 are the total number of integer solutiosn to
a+b+c = 10

- TrojanPoem

I understood !

- TrojanPoem

HAHAHAHA , I UNDERSTOOD Xd

- TrojanPoem

Give me example fast I understood xD

- dan815

this is a useful result, now u can answer questions like
say you and 2 friends have 10 dollars and only loonies allwoed now u know the total number of ways your money can be split

- TrojanPoem

You were explaining to me combinations and permutations ?

- dan815

or something with a huge solution but u can atleast get an expression for it, like
a billion dollars is in a town of 10,000 people, you can now answer the total number of ways this money can be split between the people

- dan815

the solution to his, if its loonies must be

- dan815

billion + 9999 choose 9999

- TrojanPoem

1,000,000,000 P 10,000

- dan815

not Permute but choose

- dan815

but if u want to allow cents too then
its ((billion*100 )+ 9999) choose 9999

- TrojanPoem

But Permutations are right here ?

- dan815

when we say chosoe and permute we are actually not thinking about permute and choosing

- dan815

in your mind you know the formula you get and u just say was it a choose or permute

- TrojanPoem

But how about the a+b+c+d = k ? I still can't think about it

- dan815

okay so here is what im thinking about a way to solve this

- TrojanPoem

When It was 7 you told me write them as 1111111
now 11+111+11 and calculate 2+3+2

- dan815

wait i wanna work something out on paper first

- TrojanPoem

Take your time.

- dan815

hey man I think I found something very interseting :O but im not quitee sure yet what to do with it!! but taking a look at the 2nd difference is very very very very very interseting

- dan815

I feeel like this will eventually lead to our own counting method for this whole thing

- TrojanPoem

What did you find ? :>

- dan815

okay okay look at this!! there is only 1 case we even need to lookat!

- dan815

okay for example this one
a+b+c=7
lets say
and a,b,c>= 1

- dan815

now we consider the simplst case

- dan815

1+1+5

- dan815

okay now here is what we do

- dan815

|dw:1435695479325:dw|

- dan815

now that 2nd difference can be solved with this method

- dan815

5 choose 1 ways

- dan815

so there msut be 5 solutions to this

- dan815

let see if its right!

- dan815

|dw:1435695530619:dw|

- dan815

this gives us all the solutions for fixing first number

- dan815

lets try a bigger example

- dan815

|dw:1435695642003:dw|

- dan815

i just need to deal with that first digit part lemme see about that one thing hmm

- TrojanPoem

I must give you 100 medal for that work , awesome.

- dan815

hahaha itss nothinngg

- dan815

this stuff is very interestin :D i feel like we found something very neat!

- TrojanPoem

:D

- TrojanPoem

Oh, before I forget. Dan are you good at partial integration ?

- dan815

okay wait here is the bad part

- dan815

we solved a slightly different problem

- dan815

lol because 112233 is still classified as a increasing lmao

- TrojanPoem

Oo

- dan815

but this is kind of good because we found this really cool relationship!

- dan815

ill change that question and post it ON OS, let other ppl try it!

- TrojanPoem

As you like.

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