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Combinatorics 03

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Combinatorics 03

by knight247 » Tue Sep 27, 2011 12:56 am
How many three digit odd integers less than 500 are there so that all the digits are not distinct?
A 56
B 88
C 144
D 200
E 250

Don't have an OA. Detailed explanations would be appreciated
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by chetansharma » Tue Sep 27, 2011 3:37 am
Answer is B.

If we can calculate all the odd integers below 500 with distinct digits and remove them from 250, would be the answer required.

Odd integers starting with 1 can be give as 1AB where A can take 9 values and B can take 4 values (from 1,3,5,7,9 we shd not consider 1 as we are trying to get all distinct digits)
so number of odd integers starting with 1 are 9*4 =36. The same condition holds tru for numbers starting with 3
Similarly numbers starting with 2,4 are 9*5 = 45 (respective places of A and B)

so total number of odd integers below 500 that are distinct are 36*2+ 45*2 = 162.
So the required answer = 250-162 =88

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by gmatboost » Tue Sep 27, 2011 8:36 am
There are only 200 THREE-DIGIT odd numbers below 500, so we should start with 200, rather than 250.

You are right to consider 1 and 3 separately from 2 and 4.
However, for 1 and 3, there actually
8*4 options, because once you choose the last digit, there are only 8 options left for the tens digit

And, for 2 and 4, there actually
8*5 options, because once you choose the last digit, there are only 8 options left for the tens digit

So the total with distinct is 2*8*4 + 2*8*5 = [spoiler]64 + 80 = 144. And then we subtract 200-144=56. A.[/spoiler]
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by chetansharma » Tue Sep 27, 2011 9:28 am
gmatboost wrote:There are only 200 THREE-DIGIT odd numbers below 500, so we should start with 200, rather than 250.

You are right to consider 1 and 3 separately from 2 and 4.
However, for 1 and 3, there actually
8*4 options, because once you choose the last digit, there are only 8 options left for the tens digit

And, for 2 and 4, there actually
8*5 options, because once you choose the last digit, there are only 8 options left for the tens digit

So the total with distinct is 2*8*4 + 2*8*5 = [spoiler]64 + 80 = 144. And then we subtract 200-144=56. A.[/spoiler]
Oh... what a blunder... how did I miss that!!! :shock:
Thanks for correcting me... dude

Regards,
Chetan
If my post helped you - let me know by pushing the thanks button :wink:
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