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Combinatorics

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Winning Something in the Lottery

COMBIN-YEDNN9

A lottery player selects 5 different numbers from the set $\{1, 2, \ldots, 20\}$. The lottery organizers then randomly select 5 different numbers from the same set. The player wins a prize if at least one of her numbers matches a number selected by the organizers.

What is the probability that she wins a prize?

A

$1-\left(\cfrac{3}{4}\right)^5$

B

$\cfrac{1}{4^5}$

C

$\frac{15!}{20!}$

D

$1-\cfrac{15\cdot 14\cdot 13\cdot 12\cdot 11}{20\cdot 19\cdot 18\cdot 17\cdot 16}$

E

none of the above