2020 AMC 10B Problems/Problem 19
Contents
Problem
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 ?
Solution 1
We're looking for the amount of ways we can get cards from a deck of , which is represented by .
We need to get rid of the multiples of , which will subsequently get rid of the multiples of (if we didn't, the zeroes would mess with the equation since you can't divide by 0)
, , leaves us with 17.
Converting these into, we have
~quacker88
Solution 1 but easier
We're looking for the amount of ways we can get cards from a deck of , which is represented by .
And after simplifying, we get . Now, if we examine the number , we can notice that it is equal to some number times 10. Therefore, we can divide 10 from the aforementioned expression and find the units digit, which will be .
Now, after dividing ten, we will have . We can then use modulo 10 and find that the unit digit of the expression is ~lucaswujc
Solution 2
Since this number is divisible by but not , the last digits must be divisible by but the last 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 -PCChess
Solution 3
It is not hard to check that divides the number, As , using we have . Thus , implying so the answer is .
Solution 4
As mentioned above,
We can divide both sides of by 10 to obtain
which means is simply the units digit of the left-hand side. This value is
~i_equal_tan_90, revised by emerald_block
Solution 5 (Very Factor Bashy CRT)
We note that: Let . This will help us find the last two digits modulo and modulo . It is obvious that . Also (although this not so obvious), Therefore, . Thus , implying that .
Solution 6
As in Solution 2, we see that
which contains no factors of Therefore, the sum of the digits must not be a multiple of This sum is
It follows that cannot be a multiple of ruling out choices and Therefore, our possibilities are and Now, notice that is divisible by Therefore, we can plug each possible value of into and test for divisibility by Conveniently, we see that the first value, works. Thus, the answer is (To make our argument more rigorous, we can also test divisibility by for and to show that these values do not work.)
--vaporwave
Solution 7
The total number of ways to choose from is
Using divisibility rules, we have that A is not a multiple of . Then, divide this equation by 10. This implies that the new number is divisible by but not . This means that is either or . However, is a multiple of , meaning has to be
~Arcticturn
Solution 8 (Very time consuming)
As stated in previous solutions, the number of ways to choose from is
Canceling out common factors , you get this -
When you multiply the remaining numbers, you get the product as . From this product, we can then determine that is equal to
~ KING.OF.MATH
~Puck_0 (Minor LaTeX)
Solution 9
Compute Therefore our answer is
~ Sliced_Bread
Video Solutions
Video Solution (HOW TO THINK CREATIVELY!!!)
~Education, the Study of Everything
Video Solution
Video Solution 2
https://www.youtube.com/watch?v=ApqZFuuQJ18&list=PLLCzevlMcsWNcTZEaxHe8VaccrhubDOlQ&index=6 ~ MathEx
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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