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)