21)

3 cases possible

case i) x1 > 40

so smallest value of x1 is 41

x1 + 41 + x2 + x3 = 100

x1 + x2 + x3 = 59

so possible combinations = ^{59+3-1}C_{59 }= ^{61}C_{59} = 1830

case ii) x2 > 40 and x1<=40

first we will find number of combination such that x2 > 40 and then subtract the combinations of x2 > 40 and x1 > 40

first we will find number of combination such that x2 > 40

x1 + x2 + 41 + x3 = 100

x1 + x2 + x3 = 59

so possible combinations = ^{59+3-1}C_{59 }= ^{61}C_{59} = 1830

Now number of combination such that x2 > 40 and x1 > 40

x1 + 41 + x2 + 41 + x3 = 100

x1 + x2 + x3 = 18

so possible combinations = ^{18}^{+3-1}C_{18}_{ }= ^{20}C_{18} = 190

Hence number of combination such that x2 > 40 and x1<=40 = 1830 - 190 = 1640

case iii) x3 > 40 and x1 <= 40 and x2 <= 40

first we will find number of combination such that x3 > 40 and then subtract the combinations of x3 > 40 and x1 > 40 and also subtract the combinations of x3 > 40 and x2 > 40

first we will find number of combination such that x3 > 40

x1 + x2 + x3 + 41 = 100

x1 + x2 + x3 = 59

so possible combinations = ^{59+3-1}C_{59 }= ^{61}C_{59} = 1830

Now number of combination such that x3 > 40 and x1 > 40

x1 + 41 + x2 + x3 + 41= 100

x1 + x2 + x3 = 18

so possible combinations = ^{18}^{+3-1}C_{18}_{ }= ^{20}C_{18} = 190

Now number of combination such that x3 > 40 and x2 > 40

x1 + x2 + 41 + x3 + 41= 100

x1 + x2 + x3 = 18

so possible combinations = ^{18}^{+3-1}C_{18}_{ }= ^{20}C_{18} = 190

Hence number of combination such that x3 > 40 and x1 <= 40 and x2 <= 40 = 1830 - 190 - 190 = 1450

So total ways = 1830 + 1640 + 1450 = 4920