Consider the non negative integer solutions for the equation

Consider the non negative integer solutions for the equation x_1 + x_2 + x_3 + x_4 + x_5 = 36. How many distinct solutions are there? How many distinct solutions are there if x_1 greaterthanorequalto 12? How many distinct solutions are there if x_l

Solution

Solution (a)

x1+x2+x3+x4+x5=36

The solution is given as :

The number of ways of dividing \'n\' identical objects into \'r\' groups with each group getting non-negative number of objects is given by (n+r-1) C (r-1).

n= 36 and r=5

Distinct solutions = (36+5-1) C (5-1) = 40 C 4

Solution (b)

If x1 is fixed then the number of solutions for the other variables is = 36-x1+5 C 5

where x1 is greater than equal to 12 means x1 will vary from 12 to 36

total solutions are = 29 C 5 + 28 C 5 + 27 C 5 + 26 C 5 + 25 C 5 + 24 C 5 + 23 C 5 + 22 C 5 + 21 C 5 + ..... + 5 C 5

Solution (c)

If x1 is fixed then the number of solutions for the other variables is = 36-x1+5 C 5

where x1 is less than 18 means x1 will vary from 0 to 17

total solutions are = 41 C 5 + 40 C 5 + 39 C 5 + .......... + 24 C 5