Given a set H={1,2,3,...,300. We will a create a subset of H consisting of three elements. If the sum of the three elements is divisible by 3 then the number of subsets that canbe made is x. Find the remainder if x is divided by 100000


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Question Number 187948 by horsebrand11 last updated on 24/Feb/23

$Given a set H = {1, 2, 3, . . ., 300 . We will a$ $cre a te a subset of H consisting of$ $thre e elements . If the sum of the$ $thr e e elements is divisible by 3$ $then the number of subsets that$ $canbe made is x . Find the$ $remaind e r if x is divided by 100000$
	Answered by mr W last updated on 24/Feb/23
	$H_1 ={1,4,7,..,298} with 100 elements H_2 ={2,5,8,..,299} with 100 elements H_3 ={3,6,9,..,300} with 100 elements such that the sum of 3 elements from H is divisible by 3, there are following possibilities: 1) all 3 elements are from H_3 or H_1 or H_2 2) one element from H_1 , one from H_2 and one from H_3 total number of ways: x=3×C_3 ^(100) +C_1 ^(100) ×C_1 ^(100) ×C_1 ^(100) =1 485 100 x mod 100000 =485 100$
	$H_{1} = {1, 4, 7, . ., 298} w i t h 100 e l e m e n t s$ $H_{2} = {2, 5, 8, . ., 299} w i t h 100 e l e m e n t s$ $H_{3} = {3, 6, 9, . ., 300} w i t h 100 e l e m e n t s$ $s u c h t h a t t h e s u m o f 3 e l e m e n t s f r o m$ $H i s d i v i s i b l e b y 3, t h e r e a r e f o l l o w i n g$ $p o s s i b i l i t i e s :$ $1) a l l 3 e l e m e n t s a r e f r o m H_{3} o r H_{1} o r H_{2}$ $2) o n e e l e m e n t f r o m H_{1}, o n e f r o m H_{2}$ $a n d o n e f r o m H_{3}$ $t o t a l n u m b e r o f w a y s :$ $x = 3 \times C_{3}^{100} + C_{1}^{100} \times C_{1}^{100} \times C_{1}^{100} = 1 485 100$ $x m o d 100000 = 485 100$
	Commented by cortano12 last updated on 24/Feb/23

	$sir i think it = {3 C}_{3}^{100} + 100^{3}$ $it is correct ?$
	Commented by kapoorshah last updated on 24/Feb/23

	$w r o n g$ $T h e p o s s i b i l i t i e s :$ $(1) a l l 3 e l e m e n t s a r e f r o m H_{1}$ $(2) a l l 3 e l e m e n t s a r e f r o m H_{2}$ $(3) a l l 3 e l e m e n t s a r e f r o m H_{3}$ $(4) o n e e l e m e n t f r o m H_{1}, H_{2}, H_{3} r e s p e c t i v e l y$ $t o t a l n u m b e r o f w a y s :$ $x = 3 \times C_{3}^{100} + C_{1}^{100} \times C_{1}^{100} \times C_{1}^{100}$ $= 1 485 100$
	Commented by mr W last updated on 24/Feb/23

	$y e s, 3 \times C_{3}^{100} + 100^{3} i s c o r r e c t!$
	Commented by horsebrand11 last updated on 25/Feb/23

	$Y e s . .$
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