yay thanks kamil for hint I got it
![Wink ;)](https://www.atarnotes.com/forum/Smileys/default/wink.gif)
Let the set which contains numbers that can be formed by reordering their digits be called
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
.
So for example, one type of
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
can contain
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?1,222,333,444)
or
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?1,333,222,444)
etc. This
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
contains a total of
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\frac{10!}{3!3!3!})
elements.
Another type of
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
can contain
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?1,111,122,222)
etc which would only have
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\frac{10!}{5!5!})
elements.
Therefore to count the number of multiples of
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?7)
between
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?1)
and
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?9,999,999,999)
would be denoted by
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\left \lfloor \frac{9,999,999,999}{7} \right \rfloor)
Thus using the pigeon hole theorem, if we have
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\left \lfloor \frac{9,999,999,999}{7} \right \rfloor)
many multiples of
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?7)
and
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?q)
groups of set
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
then at least one set
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
will contain
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\left \lceil \frac{\left \lfloor \frac{9,999,999,999}{7} \right \rfloor}{q} \right \rceil)
Thus if we can find out how many
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?q)
there are such that
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\left \lceil \frac{\left \lfloor \frac{9,999,999,999}{7} \right \rfloor}{q} \right \rceil \ge 10000)
then we have answered the question.
Consider using encoding to find out
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?q)
.
Let
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_i)
denote the amount of
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?i)
in a
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?10)
digit number such that
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_0+a_1+...+a_9 = 10)
So say for example, the number
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?1,222,333,444)
has
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_1 = 1)
,
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_2=a_3=a_4=3)
and the rest
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_0=a_5=a_6=...=a_9 = 0)
Now notice how
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_0+a_1+a_2+...+a_9 = 10)
Now we have created a bijection between
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_i)
and a certain set
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
which means if we can find out how many different solutions there are to
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_0+a_1+...+a_9 = 10)
then we have found out how many groups of
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\mathbb{S})
there are (which is equal to
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?q)
)
Now using a similar trick Ahmad used earlier:
Consider
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?o)
's.
That string above would represent the string
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?(a_0,a_1,...,a_9))
such that
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?(a_0,a_1,...,a_9) = (1,1...1))
Now if we move the
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?+)
sign somewhere say like this:
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?++o+o+o+o+o+o+o+ooo)
That string above would represent the string
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?= (0,0,1,1,1,1,1,1,1,3))
This means that by rearranging the
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?+)
signs we get a different set of solutions to
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a_0+a_1+a_2+...+a_9 = 10)
So how many ways can we rearrange the
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?+)
signs?
Well there's
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?19)
different 'slots' to place
and ![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?o)
and we need to pick
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?9)
slots to place a
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?+)
which means the
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?o)
slots fall naturally into place.
Therefore there are
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\binom{19}{9})
ways of rearranging the
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?+)
signs.
And since we created a bijection between the different rearrangement of the strings and
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?q)
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\therefore q = \binom{19}{9})
Subbing
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?q)
back in yields:
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?\left \lceil \frac{\left \lfloor \frac{9,999,999,999}{7} \right \rfloor}{\binom{19}{9}} \right \rceil = 15465 > 10000)
So yes there do exist at least
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?10)
digit numbers divisible by
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?7)
, all of which can be obtained from one another by a reordering of the digits