meaning of "Come down with"

[Era]

English Language Tests, Intermediate level

ESL/EFL Test #191 "Phrasal Verbs (C)", question 7

Rick ......... a terrible cold this week.

(a) came up with
(b) came about with
(c) came down with

English Language Tests, Intermediate level

ESL/EFL Test #191 "Phrasal Verbs (C)", answer 7

Rick came down with a terrible cold this week.

Correct answer: (c) came down with
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I want to explain this question.

Era

[Mister Micawber]

.
Come down with = contract (an illness)
Come up with = get an idea about; find
Come about= occur; change direction of a ship ('come about with' is not a good collocation)
.
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Native English teacher at Mister Micawber's