ellexay wrote:. 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

***

7 is the answer.

Please explain.

Let's think of the ways we can get a sum of 10:

1 9
2 8
3 7
4 6

To guarantee a sum of 10, we need to guarantee that we get a full pair.

So, we could pull:

0, 5 (not part of any pair)

then 1 from each pair (say 1, 2, 3, 4)

which gives us 6 picks without a "10".

However, on the next pick we'll definitely match a pair, so 7 is the minimum number that guarantees a pair that sum to 10.

Thinking of the least possible sum scenario....

Even if I pick 6 cards namely 0,1,2,3,4,5 the max sum of two numbers among these cards is 9. So if I pick up one more card definitely there would be a pair that adds upto 10.

So minimum cards to be picked up is 7