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Neither, since it is "qualitative data", not "quantitative". Only quantitative data can be discrete or continuous.
(Unless your assigning a numerical "value" to each nationality type, in which case it would be quantitative discrete).
Answer 2:
If considering the percentage, eg 100% Irish, 50% Irish, etc., then it is theoretically a continuous variable, assuming an infinite number of generations. However, in terms of currently existing and past living people, for any nationality, it is a discrete variable.
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