How many possibilities to arrange 0-9 in a special way (details in the body)

So I was trying to think about a problem to upload to Brilliant, and I came across a kind of difficult one that I'm not sure what's the right answer for. Try to arrange all the digits from 0-9(that's ten total) so that:

Every digit cannot be placed at the same place as its number (assuming we count places starting from 0), for example 0 can't be placed at the place 0.
As it concludes from the problem, no repetition is allowed. If I wrote 3 I can't choose it again.

Allowed arrangement: 9012345678, 1234567890

Digits representing places: 0123456789 0123456789

Explanation: As you can see, no digit in the arrangement is similar to the digit representing it place and no digit is repeated.

Forbidden arrangement: 0987654321,1123456789

Digits representing places: 0123456789 , 0 123456789

Explanation for the first number: 0 is placed at the place 0. Explanation for the second number: 1 appears more than once., 1 is located in the place 1.

The question: How many "allowed" arraignments can we make?

So my problem is that I have written a computer program using C# and I got the number 1,320,128, but the calculation I made by solving it (using combinatorics principles) brought me to 1,632,960. I prefer not to write here how exactly I got this numbers to encourage further examination and to prevent us from choosing similar ways and having similar mistakes. So I really hope at least one of my results is correct, and I can't wait to see how would you solve it! By the way, after we will confirm the solution and result I will post this problem and solution with credit to the solvers. Good luck and thank you very much!

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
list

Note: you must add a full line of space before and after lists for them to show up correctly

paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

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}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

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}`	$\boxed{123}$

Comments

Hmm, I get $1,334,961.$ Here is the relevant wiki.

Patrick Corn - 3 years, 8 months 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}`	$\boxed{123}$