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# How many 5 digit numbers are there , whose digits add up to

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Senior Manager
Joined: 21 Jun 2006
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How many 5 digit numbers are there , whose digits add up to [#permalink]  07 Aug 2006, 17:18
How many 5 digit numbers are there , whose digits add up to an odd number?
VP
Joined: 02 Jun 2006
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Edited: A correct solution is below by ps_dahiya...

Knowing odd+even =odd and odd+odd=even

Odd = {1,3,5,7,9}
Even= {2,4,6,8,0}

We have the following cases for odd sum of digits:

Case 1: 1 Odd & 5 Even
# of ways of picking 1 of 5 odd = 5
# of ways of picking 4 of 5 even = 5
Total = 25 numbers

Case 2: 3 Odd & 2 Even
# of ways of picking 3 of 5 odd = 5x4x3/3x2 = 10 ways
# of ways of picking 2 of 4 even = 4x3/2 = 6

Total number of numbers = 10x6 = 60.

Case 3: 5 Odd
# of ways of picking 5 of 5 =1

Adding all three cases= 5+1+60 = 66

Last edited by haas_mba07 on 08 Aug 2006, 04:17, edited 2 times in total.
Manager
Joined: 30 Jun 2006
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Haas_mba07,

A good way to solve the problem but did'nt you miss the numbers with 0's and numbers with repetition of digits.

Also,

Case 3: 5 Odd
# of ways of picking 5 of 5 =1

I have 5 odd digits. Then the numbers of 5 digit numbers formed by those numbers is not 1 but it should be 5! = 120. (and this is also the ones without repetition. What about numbers like 55555 or 33333 etc etc etc )

There are many other combinations which needs to be considered apart from the ones mentioned by you.
CEO
Joined: 20 Nov 2005
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Schools: Completed at SAID BUSINESS SCHOOL, OXFORD - Class of 2008
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No need of hefty calculations.
Total 5 digit numbers = 99999-9999 = 90000
Half of these digits have odd sum of digits and half them have even sum of digits.
_________________

SAID BUSINESS SCHOOL, OXFORD - MBA CLASS OF 2008

Senior Manager
Joined: 19 Jul 2006
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Total no. of 5 digits no. = 9*10*10*10*10= 90000

No. of 5-digit positive integers whose sum digits are even = No. of 5-digit positive integers whose sum digits are odd

so its 90000/2 = 45000
VP
Joined: 02 Jun 2006
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... should've thought of that!

Thanks ps/

ps_dahiya wrote:
No need of hefty calculations.
Total 5 digit numbers = 99999-9999 = 90000
Half of these digits have odd sum of digits and half them have even sum of digits.
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