# let tn denote the number of the integral sided traingles with distinct sides chosen from {1,2,3,4......n}then t20-t10=pls explain with stepwise solution

Y RAJYALAKSHMI
45 Points
9 years ago
Let the sides of triangle be a, b, c so that b + c > a
Take a = 20.
Since a, b, c are distinct -
If b = 19, c can take values from 2 to 18  = 17 distinct no. of values
if b = 18, c can take values from 3 to 17 = 15 values
simillarily when b = 17 c can take 13 values
b = 16, c can take 11 values
b = 15, c takes 9 values
b = 14, c takes 7 values
b = 13, c take 5 values
b = 12, c takes 3 values
b = 11, c takes 1 value
b can take only values up to 11, since from 10, these values get repeated
Summing up these values, we get t20 = 81

Take a = 10.
If b = 9, c can take values from 2 to 8  = 7 values
b = 8, c can take values from 3 to 7 = 5 values
b = 7 c can take 3 values
b = 6, c can take 1 value
b can take only values up to 6, since from 5, these values get repeated
Summing up these values, we get t10 = 16

So, t20 – t10 = 81 – 16 = 65