please help to break this number down 6210001000 which i understand is the answer..?

Question

There's a certain 10-digit number where that the first digit is equal to the number of zeros in the entire number, the second number is the number of 1's in the entire number, the third number is the number of 2's in the entire number, and so on, to where the 10th. digit is the number of 9's in the entire number.

Answer 1

So call the number
B = {b_0,b_1,b_2,b_3,b_4,b_5,b_6,b_7,b_8,b_9}
(this reverse indexing makes things easier).

You must have b_0 != 0, since you were told it was a ten-digit number. So b_0 is between 1 and 9.
So b_0 = 0 - (number of the b_i which are non-zero)

Also note that Sum_i=0..9 (b_i) = 10 since B has ten digits, and each b_i has to have a single value between 0..9
Hence Sum_i=1..9 (b_i) = (10 - b_0) for any particular choice of b_0

To prove the number is unique, see below:

It's easy to show 6210001000 is one possible answer.
b_0 = 6 since there are six 0's in B
b_1 = 2 since b_2, b_6 =1
b_6 = 1 since b_0 = 6
All other b_i are 0 since there are no other non-zero b_i.

To prove the number is unique, the easiest way seems to be to show that the other b_0 between 1..5 and 7..9 are impossible by (proof by contradiction).

b_0 = 9 is impossible since then b_9 must be 1, which gives a contradiction giving '9000000001', which only has 8 zeros, also b_1 must be 1, et.c

b_0 = 8 gives '8000000010', but then b_1 must be 1, and it has one less zero, so that becomes '7100000100'...
but that has two ones, so b_1++, and b_2 must be 1, and you have again one less zero so b_0=6, giving '6210001000', which finally works.

Harder to show that b_0 = 1..5 are impossible.

Let's start from the opposite extreme, b_0=1.
Then Sum_i=1..9 (b_i) = (10 - b_0) = 9
This constraint means each b_i can only be 0, 1 or 2.
Since we set b_0=1, this means we have one zero and eight ones e.g. B='11111111110', doesn't matter where you put the 0. But that doesn't work since clearly b_1 must >=8, thus Sum_i=1..9 (b_i) = 8+7>9 which is a contradiction.

It should be trivial to crank out the cases b_0=2..4, maybe there is a more compact proof.

Answer 2

6210001000 is the correct answer.
6 = number of zeroes
2 = number of ones
1 = number of twos
0 = number of threes
0 = number of fours
0 = number of fives
1 = number o sixes
0 = nuber of sevens
0 = number of eights
0= number of nines

I'm not sure why you are asking the question.

Answer 3

Isn't that Avogadro's number? 6.21 x 10^24? Sorry this probably didn't help. lol Just wondered.

Answer 4

0000000000 (I'm putting those down just for the sake of the problem)
9000000000
9000000001
8000000010

I think that that last answer satisfies the problem