Q.13 Find the number of ways in which the number 3 0 can be partitioned into three unequal parts, each
part being a natural number. What this number would be if equal parts are also included.

Ans : 61 , 75

LET THE THREE NOS BE x,y,z such that x>y>z

let x-y=k where k>0

y-z=j j>0

this implies that

x+y+z=3z+2j+k where z<28

no of solutions=coefficient of x³⁰ in (x+x²+..........x²⁷)³(x+x²=......)²(x+x²=......)

=coefficient of x³⁰ in x⁶(1-x)^-6

=coefficient of x²⁴ in (1-x)^-6

=²⁴C₅

in the above solution coefficient of x⁶ would be ²⁹C_5.

APURV PLEASE IGNORE THE ABOVE SOLUTION. IT WAS WRONG.

THE CORRECT SOLUTION IS AS FOLLOWS.

ET THE THREE NOS BE x,y,z such that x>y>z

let x-y=k where k>0

y-z=j j>0

this implies that

x+y+z=3z+2j+k where z<28

no of solutions=coefficient of x³⁰ in (x³+x⁶+..........x²⁷)(x²+x⁴......)(x+x² +......)

=coefficient of x³⁰ in x⁶(1-x³)^-1(1-x²)^-1(1-x)^-1

=61