A pure-bred normal tail, dark skin, smooth skin mouse is crossed to a pure-bred mouse that is tail-less, pale skin, and rough skin. What are the gamete genotypes produced by each of these mice?

TTppRR will produce TpR gametes.  ttPPrr  will produce tPr gametes.

Explanation:

Let us assume:

Normal tail : T, Tail less : t, Pale skin : P, Dark skin : p, Smooth skin : R, Rough skin : r

A pure-bred normal tail, dark skin, smooth skin mouse and a pure-bred mouse that is tail-less, pale skin, and rough skin  crossed together then,

TTppRR x ttPPrr

One allele from every gene comes in a gamete thus, TTppRR will produce TpR gametes.  ttPPrr  will produce tPr gametes.