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Counting with multiple criteria using data table

To be fair, the OP did provide brief snippets of data.table usage below the data dump indicating some level of effort, but posted it all in HTML (what you see we do not see), did not make the example reproducible (dput is great, and library calls really clear things up [1][2][3]), and this looks suspiciously like homework  (not on topic here, see the Posting Guide).

[1] http://stackoverflow.com/questions/5963269/how-to-make-a-great-r-reproducible-example

[2] http://adv-r.had.co.nz/Reproducibility.html

[3] https://cran.r-project.org/web/packages/reprex/index.html