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Q.13 Find the number of ways in which the number 3 0 can be partitioned into three unequal parts, eachpart 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 29C5.
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
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
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