There are four types of weights, namely 1 kg, 2 kg, 5 kg and 10 kg. What is the maximum number of different ways one can measure 20 kg, if at least eight but not more than eleven weights of 1 kg are to be used while measuring?

Let the number of 1kg weights = a (8≤a≤11), 2kg = b, 5kg = c, 10kg = d. We need a + 2b + 5c + 10d = 20. For a=8: 2b+5c+10d = 12. Solutions (b,c,d): (6,0,0),(1,2,0),(1,0,1) → 3 ways. For a=9: 2b+5c+10d = 11. Need 2b+5c+10d=11; c must be odd. c=1: 2b+10d=6 → (b,d)=(3,0),(... ); only d=0,b=3. So 1 way. For a=10: 2b+5c+10d = 10. (b,c,d): (5,0,0),(0,2,0),(0,0,1) → 3 ways. For a=11: 2b+5c+10d = 9. c=1: 2b+10d=4 → b=2,d=0. So 1 way. Total = 3+1+3+1 = 8 ways.