2 bug fix
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Japt, 78 bytes (non-competing)

This answer is non-competing because Japt was created long after this challenge.

Uo1UòJ r*U¤r*1

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How it works

Uo1UòJ r*U¤r*1   // Implicit: U = input integer                5
Uo1UòJ        // Create the inclusive range [1[-1..U)U].        [-1, 0, 1, 2, 3, 4, 5]
    r*U¤      // Slice off the first two items.             [1, 2, 3, 4, 5]
     r*1   // Reduce withby multiplication, starting at U1.   1*1=1*2=2*3=6*4=24*5=120
           // Implicit output                            120

Japt, 7 bytes (non-competing)

This answer is non-competing because Japt was created long after this challenge.

Uo1 r*U

Test it online!

How it works

Uo1 r*U   // Implicit: U = input integer
Uo1       // Create the range [1..U).
    r*U   // Reduce with multiplication, starting at U.
          // Implicit output

Japt, 8 bytes (non-competing)

This answer is non-competing because Japt was created long after this challenge.

UòJ ¤r*1

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How it works

UòJ ¤r*1   // Implicit: U = input integer                5
UòJ        // Create the inclusive range [-1..U].        [-1, 0, 1, 2, 3, 4, 5]
    ¤      // Slice off the first two items.             [1, 2, 3, 4, 5]
     r*1   // Reduce by multiplication, starting at 1.   1*1=1*2=2*3=6*4=24*5=120
           // Implicit output                            120
1
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Japt, 7 bytes (non-competing)

This answer is non-competing because Japt was created long after this challenge.

Uo1 r*U

Test it online!

How it works

Uo1 r*U   // Implicit: U = input integer
Uo1       // Create the range [1..U).
    r*U   // Reduce with multiplication, starting at U.
          // Implicit output