Sum Of All Numbers Without Repetition – Mind Your Decisions

This is a classic interview question that was once asked at Amazon.

Let S be the set of all 5 digit numbers formed by the digits 1, 2, 3, 4, 5 without repetition. What is the sum of all numbers in S?

As usual, watch the video for a solution.

Can you solve this interview question?

Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Answer To Sum Of All Numbers Without Repetition

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

We can do a smart calculation to avoid enumerating all numbers and tediously summing them.

We can sum all instances of each digit and them sum across all digits. Start with the digit 5 in the ten-thousands place.

5 _ _ _ _

The remaining 4 slots will contain the remaining 4 digits, which can be arranged in 4! ways. Thus there are 4! numbers that contain 5(10⁴), and so the sum of all instances of the digit 5 in these numbers will be 5(10⁴)4!.

But we could similarly put the digit 5 in the thousands place.

_ 5 _ _ _

By similar logic, there will be 4! numbers of this form, contributing 5(10³)4! to the sum of all numbers. There are 3 more cases to consider.

_ _ 5 _ _
_ _ _ 5 _
_ _ _ _ 5

These cases will contribute 5(10²)4!, 5(10¹)4!, and 5(10⁰)4! to the sum. The total of these values is then:

5(10⁴)4! + 5(10³)4! + 5(10²)4! + 5(10¹)4! + 5(10³)4!
= 5(10⁴ + 10³ + 10² + 10¹ + 10⁰)4!

We can do a similar calculation for the sums involving the digits 4, 3, 2, and 1.

4(10⁴ + 10³ + 10² + 10¹ + 10⁰)4!
3(10⁴ + 10³ + 10² + 10¹ + 10⁰)4!
2(10⁴ + 10³ + 10² + 10¹ + 10⁰)4!
1(10⁴ + 10³ + 10² + 10¹ + 10⁰)4!

Now we sum all of these sums. The factor (10⁴ + 10³ + 10² + 10¹ + 10⁰)4! can be factored from these to get:

(5 + 4 + 3 + 2 + 1)(10⁴ + 10³ + 10² + 10¹ + 10⁰)4!
= 15(1111)24
= 3,999,960

References

CareerCup
Amazon Interview Question for SDE1s
https://www.careercup.com/question?id=20347666

Toppr
https://www.toppr.com/ask/question/the-sum-of-all-five-digit-numbers-that-can-be-formed-using-the-digits-1234/

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PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

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