I really Need a help!

For every real positive integer $n$, define $ f(n) $ as the number of $1 $'s that appear(s) in the decimal representation of all positive integer numbers from $1$ to $n$. For example, $ f(6) =1$ and $ f(16) = 9 $. Prove that $ f(10^n) = n(10^{n-1}) + 1$ for all positive integer $n$.

#Algebra

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`2^{34}`	$2^{34}$
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Comments

There is a simple combinatorial argument for this.

Notice that $f(n)=k+f(n-1)$ where $k$ is the number of 1s in $n$ . Now, we have only a single 1 in powers of 10, so $f(10^n)=f(10^n-1)+1$ . Now, all numbers $\leq 10^n-1$ are at most $n$ digit numbers (can be thought of as $n$ digit numbers using non negative digits). Now, we need to count the numbers having 1 in their digits. Fixing 1 at one of the digits places is done in $n$ ways and the other $n-1$ digit places can be assigned a digit from 0 to 9 in 10 ways for each digit place, hence $10^{n-1}$ ways. Notice that there's no problem with excessive counting or missing out on counting a few since if a number has $m$ 1s, it is counted exactly once for fixing 1 at each of the $m$ digit places, thus it getting counted exactly $m$ times as desired. Now then, by the rule of product, we have $f(10^n-1)=n10^{n-1}$ and hence we conclude that,

$f(10^n)=n\cdot 10^{n-1}+1$

Prasun Biswas - 4 years ago

Can you give a proof for $f(n) = k + f(n-1)$ ? I still don't get it..

Fidel Simanjuntak - 4 years ago

If $f(n)$ is the number of 1s in the numbers from 1 to $n$ , then this is equivalent to the number of 1s in the numbers from 1 to $n-1$ plus the number of 1s in $n$ , right? Denoting by $k$ the number of 1s in $n$ , we can write $f(n)=k+f(n-1)$ .

Prasun Biswas - 4 years ago

@Prasun Biswas – What do you mean, by saying "fixing" ?

Fidel Simanjuntak - 4 years ago

@Fidel Simanjuntak – By "fixing", I mean specifying one of the digit places to be 1 (hence making the digit place fixed) and letting the other digit places vary.

Such "fixing" argument tend to be common in counting (enumeration) problems.

Prasun Biswas - 4 years ago

@Prasun Biswas – I'm sorry for bothering you, but I still don't get this " Notice that there's no problem with excessive counting or ........". Can you help me?

Fidel Simanjuntak - 4 years ago

@Fidel Simanjuntak – That part was to show that if a number has $m$ 1s, it is getting counting exactly $m$ times, no more no less. This is because we want to count the number of occurrences of 1s for that number, which is $m$ .

And don't worry, if you have any further problems understanding my solution, feel free to ask. I'll try my best to help you understand. After all, a solution is no good if it can't be completely understood by the reader. ;)

PS: I'm typing from my mobile, so I might be late responding to your queries, sorry for that.

Prasun Biswas - 4 years ago

What do you mean, by saying "fixing" ?

Fidel Simanjuntak - 4 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$