Question
Total number of non-negative integral solutions of 2x + y + z = 21 is
You can solve this question by the following method
possile algebric expression for the given equation is
(x0+x2+x4+........+x20)(x0+x1+x2+..................+x21)(x0+x1+x2+x3+....................x21) (i)
Here three brackets are for x , y and z, and powers are based on the range of least to greatest possile values of x, y, z
on solving eq (i) will become (x0+x2+x4+...........+x20)(1-x22)2(1-x)-2
now we have to calculate coefficient of x21 in this expression so (1-x22)2 can be omitted
general term of (1-x)-2 is (r+1)xr
so required coefficient = 22+20+18+16+14+12+10+8+6+4+2 = 132
You can solve this question by the following method
possile algebric expression for the given equation is
(x0+x2+x4+........+x20)(x0+x1+x2+..................+x21)(x0+x1+x2+x3+....................x21) (i)
Here three brackets are for x , y and z, and powers are based on the range of least to greatest possile values of x, y, z
on solving eq (i) will become (x0+x2+x4+...........+x20)(1-x22)2(1-x)-2
now we have to calculate coefficient of x21 in this expression so (1-x22)2 can be omitted
general term of (1-x)-2 is (r+1)xr
so required coefficient = 22+20+18+16+14+12+10+8+6+4+2 = 132
Read 1 Solution.
write it as , x+x+y+z=21
here 0 can be included so number of values are 24C3/2 .