Problem:
In a certain card game, a player is dealt a hand of 10 cards from a deck of 52 distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as 158A00A4AA0. What is the digit A?
Answer Choices:
A. 2
B. 3
C. 4
D. 6
E. 7
Solution:
We're looking for the number of ways we can get 10 cards from a deck of 52, which is represented by (1052).
(1052)=10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅152⋅51⋅50⋅49⋅48⋅47⋅46⋅45⋅44⋅43
And after simplifying, we get
26⋅17⋅7⋅47⋅46⋅5⋅11⋅43
Now, if we examine the number
158A00A4AA0
we can notice that it is equal to some number n times 10 . Therefore, we can divide 10 from the aforementioned expression and find the unit digit, which will be A.
Now, after dividing ten, we will have
26⋅17⋅7⋅47⋅23⋅11⋅43
We can then use modulo 10 and find that the unit digit of the expression is (A)2.
OR
(1052)=10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅152⋅51⋅50⋅49⋅48⋅47⋅46⋅45⋅44⋅43=26⋅17⋅5⋅7⋅47⋅46⋅11⋅43
Since this number is divisible by 4 but not 8, the last 2 digits must be divisible by 4 but the last 3 digits cannot be divisible by 8. This narrows the options down to 2 and 6.
Also, the number cannot be divisible by 3. Adding up the digits, we get 18+4A. If A=6, then the expression equals 42, a multiple of 3. This would mean that the entire number would be divisible by 3, which is not what we want
Therefore, the only option is (A)2.
The problems on this page are the property of the MAA's American Mathematics Competitions