Maths problem in desciption any help is needed. I appreciate all who choose to.

Question

Zaina is curious about numbers that contain the number 2016. She insists that the digits 0, 1, 2, 6 occur only once in the number and they appear as the uninterrupted sequence 2016. She calls such a number a Year Number. For example, 5720169 and 3201673 are year numbers but 5207169 and 18201653 are not.

Question 1:Starting with the smallest, write down the first ten year numbers in numerical order.

Question 2:Find the 2016th year number.

Answer 1

The first thing you should do with a problem like this is to read and re-read it until you understand it and can put it into your own words.

Here's mine: This problem reserves the digits 2,0,1,6 for a single use. They appear in order and are not separated (that’s called contiguous). We could call that portion of a larger number "X." Note that 0 is already used, so we don't have to worry about leading zeros; we can have leading 3,4,5,7,8,9 and trailing 3,4,5,7,8,9. The problem does not specifically state whether we are allowed to use those digits multiple times, but it says that "3X73" is a year number and that 2,0,1,6 have single use.

Now, we have to determine how to produce numbers in ascending order with X=2016 internal to the number.
The first year number has 4 digits: 2016.

The next few year numbers will have 5 digits, so we must sort 20163, 20164, 20165, 20167, 20168, 20169, and 32016, 42016, 52016, 72016, 82016, 92016 into ascending order. Hey! 0,1,2 are already used so these are in order. Also, the problem asked for the first 10, and there 12 more here, so the first 10 year numbers are:
2016, 20163, 20164, 20165, 20167, 20168, 20169, 32016, 42016, 52016

So far, we have 4- and 5-digit numbers containing “X” and 3,4,5,7,8,9.

Remember, in decimal, we have 10 digits (0-9) and with a positional number system, two digits form 100 possible values (00-99). That is 10*10. Three digits form 1000 possible values (000-999).

So, with 6 digits available, using one digit after ”X” made 6 numbers and using one digit before “X” make 6 more numbers, That was 12.

Using no digits: 1 number
Using 1 digit: 2*6 = 12 numbers [and X3 is the lowest]
Using 2 digits: 3*36=108 numbers [and X34 is the lowest]
Note: there are 6*6=36 Xnn values, 6*6=36 nXn values, and 6*6=36 nnX values. All of these are less than any value using 3 digits.
So far, we have 1+12+108=121 numbers. Keep going until we get the 2016th number.

This looks like: (k+1)*6^k for k=0,1,2,3… each time (1, 12, 108, 864, 6480, …

We want to find the 2016th; it uses 4 other digits (the nnnn).
    2016-1-12-108-864 = 1031 more.
So, 2106th Year Number uses 4 digits with an X.
    4*6*6*6 = 864
So far: X7nnn
    [note that “7” is the fourth digit of 3,4,5,7,8,9].
We still need 2016-1-12-108-864-864 = 167
    4*6*6 = 144.
We have X77nn [note: 7 is fourth]
Now, we need 2016-1-12-108-864-864-144 = 23 more.
    3*6 = 18.
We have X775n [note: 5 is third possible digit]
Now, we need 2016-1-12-108-864-864-144-18 = 5 more.
We have X7758 [note: 8 is fifth possible digit]

No more digits are needed, so 20167758 is the 2016th Year Number.

Answer 2

Answer removed by Author.