We proceed directly.
60 ! 31 ! ≡ ( 60 ) ( 59 ) . . . ( 32 ) mod 31 ≡ ( 29 ) ( 28 ) . . . ( 1 ) mod 31 ≡ 29 ! mod 31 ≡ 30 ! ⋅ 30 − 1 mod 31 ≡ ( 31 − 1 ) ! ⋅ ( − 1 ) mod 31 ≡ ( − 1 ) ( − 1 ) mod 31 ≡ 1 mod 31 {\displaystyle \displaystyle {\begin{aligned}{\frac {60!}{31!}}&\equiv (60)(59)...(32)\mod {31}\\&\equiv (29)(28)...(1)\mod {31}\\&\equiv 29!\mod {31}\\&\equiv 30!\cdot 30^{-1}\mod {31}\\&\equiv (31-1)!\cdot (-1)\mod {31}\\&\equiv (-1)(-1)\mod {31}\\&\equiv 1\mod {31}\end{aligned}}}
where in the second last line we used Wilson's Theorem. We also used the fact that
30 − 1 ≡ ( − 1 ) − 1 ≡ − 1 mod 31 {\displaystyle \displaystyle 30^{-1}\equiv (-1)^{-1}\equiv -1\mod {31}} .
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