A Card Game

Probability Level 3

I have a bag of cards numbered from 1 to 100 while my friend has a bag of cards numbered from 1 to 200. The goal is for me to successively pick two cards without replacement from my bag, such that the sum is equal to the single card that my friend picks from his bag. For example, if my friend picks up the card 4, then I must pick up $(1,3)$ or $(3,1)$ . I cannot pick up $(2,2)$ .

Over all card choices, how many ways can the goal be met?

Note: Order does matter.

The answer is 9900.

2 solutions

A Former Brilliant Member
Feb 16, 2016

No matter what cards I choose, my friend would always have 1 way to choose the card that is the sum of mine. Thus, we simply need to find how many ways I can select two cards, which is 99*100=9900.

Arulx Z
Feb 13, 2016

Writing all the possibilities is very tedious so we look for a better way.

It's important to notice that maximum sum of my cards will be equal to $100 + 99 = 199$ . Since my friend has cards numbered from 1 to 200, every time I choose any two cards, my friend will always have a card with sum equal to my cards.

So we just need to find the number of ways in which I can choose two cards, which is $100 \cdot 99 = 9900$ .

Moderator note:

Having the right perspective makes this problem much easier to approach. If we started thinking about "For each card that my friend has, how many ways can I ...".

If the problem was written up in a way to reflect this, it would be much better :)

But you cannot express 1 and 2 as a sum of 2 distinct numbers. So you are left with 197 numbers which can be expressed as the sum of two distinct numbers [1,100] For example if you pull out any two numbers and your friend pulls out either 1 or 2 then this becomes an unwanted case

Surya Sharma - 5 years, 4 months ago

According to my solution, my friend can only pull a card whose number is sum of my cards. So it's impossible for the friend to pull 1 or 2.

Arulx Z - 5 years, 4 months ago

@Calvin Lin I have modified the question accordingly.

Arulx Z - 5 years, 3 months ago

The phrasing "For each card my friend picks up" is wrong, because it means "If my friend picks up the card 1, ...". This is not what you want.

Calvin Lin Staff - 5 years, 3 months ago

I have modified the question again. I'm a bit confused so please check it again for ambiguity. Thanks :)

Arulx Z - 5 years, 3 months ago

@Arulx Z – Same issue. "For each card my friend has, in how many ways can I do X" means "If my friend picks up a specific card (say 1), how many ways can I do X" and not "Find the sum of the number of ways I can do X across all possible values of N".

I have edited the problem accordingly.

Calvin Lin Staff - 5 years, 3 months ago

@Calvin Lin – Thanks a lot

Arulx Z - 5 years, 3 months ago

A Card Game

The answer is 9900.

2 solutions

Moderator note:

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