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?


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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

	$M E T H O D I$ $s a y t h e l e n g t h e s o f t h e p a r t s a r e$ $a, b, c w i t h a ⩽ b ⩽ c .$ $a + b + c = 100$ $100 = a + b + c ⩾ 3 a$ $\Rightarrow a ⩽ \frac{100}{3} \Rightarrow a ⩽ 33$ $w i t h a = 1 :$ $99 = b + c ⩾ 2 b$ $\Rightarrow b ⩽ \frac{99}{2} \Rightarrow b ⩽ 49$ $\Rightarrow b = 1, 2, . . ., 49 \Rightarrow 49 s o l u t i o n s$ $w i t h a = 2 :$ $98 = b + c ⩾ 2 b$ $\Rightarrow b ⩽ 49$ $\Rightarrow b = 2, 3, . . ., 49 \Rightarrow 48 s o l u t i o n s$ $w i t h a = 3 :$ $97 = b + c ⩾ 2 b$ $\Rightarrow b ⩽ \frac{97}{2} \Rightarrow b ⩽ 48$ $\Rightarrow b = 3, 4, . . ., 48 \Rightarrow 46 s o l u t i o n s$ $w i t h a = 4 :$ $96 = b + c ⩾ 2 b$ $\Rightarrow b ⩽ 48$ $\Rightarrow b = 4, 5, . . ., 48 \Rightarrow 45 s o l u t i o n s$ $. . . . .$ $w i t h a = 33 :$ $67 = b + c ⩾ 2 b$ $\Rightarrow b ⩽ \frac{67}{2} \Rightarrow b ⩽ 33$ $\Rightarrow b = 33 \Rightarrow 1 s o l u t i o n$ $t o t a l n u m b e r o f s o l u t i o n s :$ $1 + 4 + 7 + . . . + 46 + 49$ $+ 3 + 6 + 9 + . . . + 45 + 48$ $= \frac{17 \times (1 + 49)}{2} + \frac{16 \times (3 + 48)}{2}$ $= 833$
	Commented by bobhans last updated on 18/Jul/20

	$c o o l l . . . . n i c e$
	Commented by mr W last updated on 18/Jul/20

	$M E T H O D I I$ $s a y t h e l e n g t h e s o f t h e p a r t s a r e$ $a, b, c w i t h a ⩽ b ⩽ c .$ $a + b + c = 100 . . . (i)$ $n u m b e r o f s o l u t i o n s i s t h e c o e f f . o f$ $x^{100} t e r m o f f o l l o w i n g g e n e r a t i n g$ $f u n c t i o n$ $\frac{x^{3}}{(1 - x) (1 - x^{2}) (1 - x^{3})}$ $w h i c h i s 833 .$
	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

	$y e s$
	Commented by bemath last updated on 18/Jul/20

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