2020 AMC 10 B
Complete problem set with solutions and individual problem pages
In a certain card game, a player is dealt a hand of cards from a deck of distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as . What is the digit ?(2020 AMC 10B, Question #19)
- A.
- B.
- C.
- D.
- E.
Solution 1:
We're looking for the amount of ways we can get 10 cards from a deck of 52 , which is represented by . We need to get rid of the multiples of 3 , which will subsequently get rid of the multiples of 9 (if we didn't, the zeroes would mess with the equation since you can't divide by 0 ) leaves us with . Converting these into , we have
Solution 2:
Since this number is divisible by but not , the last digits must be divisible by but the last 3 digits cannot be divisible by . This narrows the options down to and . Also, the number cannot be divisible by . Adding up the digits, we get . If , then the expression equals , a multiple of . This would mean that the entire number would be divisible by , which is not what we want. Therefore, the only option is (A)2.
