Question Number 103903 by mr W last updated on 18/Jul/20

a 100 cm long rod should be divided  into 3 parts. the length of each part  in cm should be integer. in how   many different ways can this be  done?

Commented bybobhans last updated on 18/Jul/20

in parts alowed the same length?

Commented bymr W last updated on 18/Jul/20

the only condition is: the length  must be integer. so the parts may  also have equal length.

Commented bybobhans last updated on 18/Jul/20

it same x_1 +x_2 +x_3  = 100 , with x_i ∈Z^+

Commented bymr W last updated on 18/Jul/20

not exactly the same.   e.g. 2+5+93 or 5+2+93 or 93+2+5 are  different solutions for your  equation, but they represent the one  and the same way to divide the rod.  the parts are not labeled.

Commented bybobhans last updated on 18/Jul/20

o oo yes agree

Answered by mr W last updated on 18/Jul/20

METHOD I  say the lengthes of the parts are  a,b,c with a≤b≤c.  a+b+c=100  100=a+b+c≥3a  ⇒a≤((100)/3) ⇒a≤33  with a=1:  99=b+c≥2b  ⇒b≤((99)/2) ⇒b≤49  ⇒b=1,2,...,49 ⇒49 solutions  with a=2:  98=b+c≥2b  ⇒b≤49  ⇒b=2,3,...,49 ⇒48 solutions  with a=3:  97=b+c≥2b  ⇒b≤((97)/2) ⇒b≤48  ⇒b=3,4,...,48 ⇒46 solutions  with a=4:  96=b+c≥2b  ⇒b≤48  ⇒b=4,5,...,48 ⇒45 solutions  .....  with a=33:  67=b+c≥2b  ⇒b≤((67)/2) ⇒b≤33  ⇒b=33 ⇒1 solution    total number of solutions:  1+4+7+...+46+49  +3+6+9+...+45+48  =((17×(1+49))/2)+((16×(3+48))/2)  =833

Commented bybobhans last updated on 18/Jul/20

cooll....nice

Commented bymr W last updated on 18/Jul/20

METHOD II  say the lengthes of the parts are  a,b,c with a≤b≤c.  a+b+c=100   ...(i)  number of solutions is the coeff. of  x^(100)  term of following generating  function  (x^3 /((1−x)(1−x^2 )(1−x^3 )))  which is 833.

Commented bymr W last updated on 18/Jul/20

no sir!  (a+b+c)^(100)  has no meaning.  it is the coefficient of x^(100)  in the  generating function  (x^3 /((1−x)(1−x^2 )(1−x^3 )))

Commented bymr W last updated on 18/Jul/20

Commented bybemath last updated on 18/Jul/20

coeff of x^(100)  from (a+b+c)^(100)   sir?

Commented bymr W last updated on 18/Jul/20

it means in how many ways can the  number 100 be divided into 3 parts.

Commented bybemath last updated on 18/Jul/20

if number 200 be divided  into 3 parts then equal to  coeff of x^(200)  sir?

Commented bymr W last updated on 18/Jul/20

yes

Commented bybemath last updated on 18/Jul/20

thank you sir. i will learn

Commented byprakash jain last updated on 18/Jul/20

Some more ideas around this topic in Wikipedia https://en.m.wikipedia.org/wiki/Partition_(number_theory)