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-

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

a 100 c m l o n g r o d s h o u l d b e d i v i d e d

i n t o 3 p a r t s . t h e l e n g t h o f e a c h p a r t

i n c m s h o u l d b e i n t e g e r . i n h o w

m a n y d i f f e r e n t w a y s c a n t h i s b e

d o n e ?

Commented by bobhans last updated on 18/Jul/20

i n p a r t s a l o w e d t h e s a m e l e n g t h ?

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

t h e o n l y c o n d i t i o n i s : t h e l e n g t h

m u s t b e i n t e g e r . s o t h e p a r t s m a y

a l s o h a v e e q u a l l e n g t h .

Commented by bobhans last updated on 18/Jul/20

i t s a m e x_{1} + x_{2} + x_{3} = 100, w i t h x_{i} \in Z^{+}

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

n o t e x a c t l y t h e s a m e .

e . g . 2 + 5 + 93 o r 5 + 2 + 93 o r 93 + 2 + 5 a r e

d i f f e r e n t s o l u t i o n s f o r y o u r

e q u a t i o n, b u t t h e y r e p r e s e n t t h e o n e

a n d t h e s a m e w a y t o d i v i d e t h e r o d .

t h e p a r t s a r e n o t l a b e l e d .

Commented by bobhans last updated on 18/Jul/20

o o o y e s a g r e e

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

\boldsymbol M E T H O D \boldsymbol I

s a y t h e l e n g t h e s o f t h e p a r t s a r e

a, b, c w i t h a ⩽ b ⩽ c .

a + b + c = 100

100 = a + b + c ⩾ 3 a

\Rightarrow a ⩽ \frac{100}{3} \Rightarrow a ⩽ 33

w i t h a = 1 :

99 = b + c ⩾ 2 b

\Rightarrow b ⩽ \frac{99}{2} \Rightarrow b ⩽ 49

\Rightarrow b = 1, 2, \dots, 49 \Rightarrow 49 s o l u t i o n s

w i t h a = 2 :

98 = b + c ⩾ 2 b

\Rightarrow b ⩽ 49

\Rightarrow b = 2, 3, \dots, 49 \Rightarrow 48 s o l u t i o n s

w i t h a = 3 :

97 = b + c ⩾ 2 b

\Rightarrow b ⩽ \frac{97}{2} \Rightarrow b ⩽ 48

\Rightarrow b = 3, 4, \dots, 48 \Rightarrow 46 s o l u t i o n s

w i t h a = 4 :

96 = b + c ⩾ 2 b

\Rightarrow b ⩽ 48

\Rightarrow b = 4, 5, \dots, 48 \Rightarrow 45 s o l u t i o n s

\dots . .

w i t h a = 33 :

67 = b + c ⩾ 2 b

\Rightarrow b ⩽ \frac{67}{2} \Rightarrow b ⩽ 33

\Rightarrow b = 33 \Rightarrow 1 s o l u t i o n

t o t a l n u m b e r o f s o l u t i o n s :

1 + 4 + 7 + \dots + 46 + 49

+ 3 + 6 + 9 + \dots + 45 + 48

= \frac{17 \times (1 + 49)}{2} + \frac{16 \times (3 + 48)}{2}

= 833

Commented by bobhans last updated on 18/Jul/20

c o o l l \dots . n i c e

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

\boldsymbol M E T H O D \boldsymbol I I

s a y t h e l e n g t h e s o f t h e p a r t s a r e

a, b, c w i t h a ⩽ b ⩽ c .

\boldsymbol a + \boldsymbol b + \boldsymbol c = 100 \dots (i)

n u m b e r o f s o l u t i o n s i s t h e c o e f f . o f

x^{100} t e r m o f f o l l o w i n g g e n e r a t i n g

f u n c t i o n

\frac{x^{3}}{(1 - x) (1 - x^{2}) (1 - x^{3})}

w h i c h i s 833 .

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

n o s i r!

{(a + b + c)}^{100} h a s n o m e a n i n g .

i t i s t h e c o e f f i c i e n t o f x^{100} i n t h e

g e n e r a t i n g f u n c t i o n

\frac{x^{3}}{(1 - x) (1 - x^{2}) (1 - x^{3})}

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

Commented by bemath last updated on 18/Jul/20

c o e f f o f x^{100} f r o m {(a + b + c)}^{100}

s i r ?

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

i t m e a n s i n h o w m a n y w a y s c a n t h e

n u m b e r 100 b e d i v i d e d i n t o 3 p a r t s .

Commented by bemath last updated on 18/Jul/20

i f n u m b e r 200 b e d i v i d e d

i n t o 3 p a r t s t h e n e q u a l t o

c o e f f o f x^{200} s i r ?

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

y e s

y e s

Commented by bemath last updated on 18/Jul/20

t h a n k y o u s i r . i w i l l l e a r n

t h a n k y o u s i r . i w i l l l e a r n

Commented by prakash jain last updated on 18/Jul/20

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