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Each of the integers from 0 to 9, inclusive, is written on a [#permalink]
07 May 2005, 00:06

Each of the integers from 0 to 9, inclusive, is written on a separate slip of blank paper and the ten slips are dropped into a hat. If the slips are then drawn one at a time without replacement, how many must be drawn to ensure that the numbers on two of the slips drawn will have a sum of 10?

3

4

5

6

7

--------------

I did get the ans right, but I hoping there's a better method than the one I used.

with the integers from 0 to 9, you can add 10 with
1+9
2+8
3+7
4+6
other numbers are 0,5

we should find a combination of 7 6 5 4 or 3 that does not add 10.
I started from 5 -> 9 8 7 6 5 (pick one number from first/second column and 0 or 5)
now 6-> pick one number from first/second column and both 0 and 5
0 5 9 8 7 6 or 0 5 1 2 3 4
7 is the answer because you pick
0 and 5 first
6 7 8 9 OR 1 2 3 4
next number will necessarily lead to 10 as a sum

You have to just assume the worst - that you won't get 10 until it's absolutely impossible to get anything else.

So what makes ten?

1-9
2-8
3-7
4-6

That's it. So what if the first number you choose is 0? You'll never get ten. What if the next one is 5? Same problem. So now you've chosen 2 numbers already, and no 10.

Then, what if you pick all the next numbers that don't have a partner? Let's say you pick 1,2,3,4 in a row. None of your number now will add up to 10, and you've chosen 6 numbers.

What's left? The 4 partners to 1,2,3,4 - so the next number you choose, whatever it is, will definately match with one of them to make 10.

Agree with you guys. You have to skip 0 and 5 whixh do not lead to any correct sum. Then the 4 first component of this "10" sum. The following one must at least complete one of the possible sums.

There are 6 numbers (0,1,2,3,4,5) forming a set in which any two numbers do not add up to 10.

So, in the worst case scenario (if you were to be absolutely sure that there are 2 numbers that add to 10), You need to pick these 6 numbers + one more. So, the answer is 7.