Difference between revisions of "2020 AMC 10B Problems/Problem 19"
Emathmaster (talk | contribs) (→Solution 2) |
Emathmaster (talk | contribs) (→Solution 3) |
||
Line 33: | Line 33: | ||
==Solution 3== | ==Solution 3== | ||
+ | It is not hard to check that <math>7</math> divides the number. As <math>10^3\equiv-1\pmod{7}</math>, using <math>\pmod{7}</math> we have <math>7|\overline{}</math>. | ||
==Video Solution== | ==Video Solution== |
Revision as of 20:21, 7 February 2020
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 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
.
Video Solution
~IceMatrix
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.