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Problem of the Week

Problem E and Solution

Such a Card


Johanna has a deck of cards with the following properties:

  1. Each card in the deck has a positive three-digit integer on it.

  2. There is exactly one card in the deck for every three-digit positive integer.

Johanna randomly selects a card from the deck of cards. Determine the probability that the sum of the digits of the integer on this card is 15.

Card numbered 456.


To begin, we need to determine the number of cards in the deck. Since there is a card for each three-digit positive integer, there are 900 cards in the deck. We must be careful calculating this number. There are 999 positive integers less than 1000. Of this set, 90 are two-digit numbers and 9 are single-digit numbers. Therefore there are \(999-90-9=900\) three-digit positive integers.
Next we need to determine the digit combinations on a card that have a sum of 15. We will determine the possibilities using cases. Then we will look at the specific groups of numbers that sum to 15 to count the number of cards produced from each group.

Now that we have the groups of numbers, we can determine the number of cards that can be created from each group of three numbers. We will do this again with cases: groups containing a 0, groups containing three distinct numbers but not 0, groups containing exactly two numbers the same but not 0, and groups containing three numbers the same but not 0.

Combining the counts from the above four cases, there are \(8+48+12+1=69\) cards in the deck with a digit sum of 15. Therefore, the probability that Johanna selects card whose digits add to 15 is \(\dfrac{69}{900}=\dfrac{23}{300}\). This translates to approximately a \(7.7\%\) chance.
A game is considered fair if there is close to a 50% chance of winning. If Johanna was playing a game where she can win by drawing a card whose digits sum to 15, then this game is definitely not fair. If you changed the game to “if the card chosen has a sum that is divisible by 5”, there is about a 20% chance of winning. This is better but still not fair.
Can you create a game using this specific deck of cards that is reasonably fair and fun to play?