Answer: 8

Let $a, b, c, d$ be the number of 1, 2, 5, 10 kg weights. We need $a + 2b + 5c + 10d = 20, \quad a \in \{8, 9, 10, 11\},\ b,c,d \ge 0.$

Case $a = 8$: $2b + 5c + 10d = 12$

• $d=0$: $(b,c) = (6,0), (1,2)$ → 2 ways
• $d=1$: $(b,c) = (1,0)$ → 1 way

Subtotal: 3

Case $a = 9$: $2b + 5c + 10d = 11$ (odd, so $c$ must be odd)

• $d=0$: $(b,c) = (3,1)$ → 1 way

Subtotal: 1

Case $a = 10$: $2b + 5c + 10d = 10$

• $d=0$: $(b,c) = (5,0), (0,2)$ → 2 ways
• $d=1$: $(b,c) = (0,0)$ → 1 way

Subtotal: 3

Case $a = 11$: $2b + 5c + 10d = 9$ ($c$ odd)

• $d=0$: $(b,c) = (2,1)$ → 1 way

Subtotal: 1

Total: $3 + 1 + 3 + 1 = 8$