How many five-digit numbers are there who digits sum to 39?

Can someone give me their answer so I can check if mine is right? Thank you!

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- Jan 29th 2011, 05:38 AMCuriosityCabinetCombinatorics question involving summing digits of numbers
How many five-digit numbers are there who digits sum to 39?

Can someone give me their answer so I can check if mine is right? Thank you! - Jan 29th 2011, 05:52 AMPlato
- Jan 29th 2011, 06:00 AMCuriosityCabinet
The different digit combinations possible are of the form:

99993

99984

99975

99885

98886

99966

88887

99876

99777

98877

Then I worked out how many combinations of each digit there are and summed it, to get 210. But have a feeling that isn't quite right. - Jan 29th 2011, 06:06 AMPlato
- Jan 29th 2011, 06:12 AMCuriosityCabinet
- Jan 29th 2011, 06:21 AMPlato
One expands $\displaystyle \left( {\sum\limits_{k = 1}^9 {x^k } } \right)\left( {\sum\limits_{k = 0}^9 {x^k } } \right)^4 $.

The coefficient of $\displaystyle x^{39}$ is the answer.

Notice the first sum starts at $\displaystyle k=1$ because the first digit in a five digit number cannot be zero.

Here is an online tool you can use.