SOLUTION: Ten-digit number. Find a 10-digit number whose first digit is the number of 1�s in the 10-digit number, whose second digit is the number of 2�s in the 10-digit number, whose thi

Question 398690: Ten-digit number. Find a 10-digit number whose first digit
is the number of 1�s in the 10-digit number, whose second
digit is the number of 2�s in the 10-digit number, whose
third digit is the number of 3�s in the 10-digit number, and
so on. The ninth digit must be the number of nines in the
10-digit number and the tenth digit must be the number of
zeros in the 10-digit number
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
I'll denote the digits , , ..., . Since there are 1's, 2's, ..., 0's, then because there are ten digits in the number.

Suppose . Then a contradiction would result, since there is at least one zero, so .

Suppose . Then, this implies that there is exactly one zero within the number, and all the other 's are at least one, which also creates a contradiction due to the sum of digits. This can also apply to higher values of . If , then . Also, if n of the numbers { , , ..., } are zero, then 9-n of these numbers are greater than or equal to 1, and they sum up to 10-n.

By the Pigeonhole principle, exactly one of the 9-n numbers is equal to 2 and all other nonzero numbers are equal to 1 (this means, 8-n numbers equal to 1). Therefore the number has n zeros, 8-n 1's and one 2. We can easily guess and check based on the value of n, which can only range between 2 and 8. We see that the number 2100010006 satisfies all the given constraints, which happens when .
RELATED QUESTIONS
find a 4 digit number whose first digit is 2 .the second digit is 1 more than the first... (answered by ikleyn)

one third of a three-digit number whose units digit is x, ten digit is 5 and hundreds... (answered by edjones,ankor@dixie-net.com)

The untis digit of a two-digit number exceeds twice the tens digit by 1. Find the number... (answered by richard1234)

The sum of the digits of a 2 digit number is 6. The second digit is twice the first... (answered by Edwin McCravy)

Can you help find the answer? a four digit odd number whose ones digit is half the... (answered by Alan3354)

In a three digit number, the sum of its ones digit and tens digit is ten. Its tens digit... (answered by Edwin McCravy)

A number whose tens digit is twice its units... (answered by stanbon)

An even 2 digit number whose 2 digits have a difference of 1 and the first digit is... (answered by stanbon)

Find the number of 10-digit numbers whose product of digits is... (answered by Edwin McCravy)