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Question Number 47656 by Necxx last updated on 12/Nov/18

A square is divided into 9 identical  smaller squares.Six identical balls  are to be placed in these smaller   squares such that each of the three  rows gets at least one ball(one  ball in one square only).In how  many different ways can this be  done?  a)91 b)51 c)81 d)41

$${A}\:{square}\:{is}\:{divided}\:{into}\:\mathrm{9}\:{identical} \\ $$$${smaller}\:{squares}.{Six}\:{identical}\:{balls} \\ $$$${are}\:{to}\:{be}\:{placed}\:{in}\:{these}\:{smaller}\: \\ $$$${squares}\:{such}\:{that}\:{each}\:{of}\:{the}\:{three} \\ $$$${rows}\:{gets}\:{at}\:{least}\:{one}\:{ball}\left({one}\right. \\ $$$$\left.{ball}\:{in}\:{one}\:{square}\:{only}\right).{In}\:{how} \\ $$$${many}\:{different}\:{ways}\:{can}\:{this}\:{be} \\ $$$${done}? \\ $$$$\left.{a}\left.\right)\left.\mathrm{9}\left.\mathrm{1}\:{b}\right)\mathrm{51}\:{c}\right)\mathrm{81}\:{d}\right)\mathrm{41} \\ $$$$ \\ $$

Commented by Necxx last updated on 13/Nov/18

please help

$${please}\:{help} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Nov/18

combination...let denote  first row=FR   similarly SR,   TR  1)FR=1  SR=2  TR=3→9ways  2)FR=2  SR=3  TR=1→9ways  3)FR=3  SR=2 TR=1→9ways  4)FR=2  SR=2  TR=2  →27ways  5)FR= 3   SR=1   TR=2  →9ways  6)FR=2   SR=1   TR=3  →9ways  7)FR=1  SR=3   TR=2→ 9ways    (1)3c_1 ×3c_2 ×3c_3 =9 ←example of calculation for  first case that is FR=1  SR=2  TR=3  others similar way calculated)  so total 9×6+27=81

$${combination}...{let}\:{denote} \\ $$$${first}\:{row}={FR}\:\:\:{similarly}\:{SR},\:\:\:{TR} \\ $$$$\left.\mathrm{1}\right){FR}=\mathrm{1}\:\:{SR}=\mathrm{2}\:\:{TR}=\mathrm{3}\rightarrow\mathrm{9}{ways} \\ $$$$\left.\mathrm{2}\right){FR}=\mathrm{2}\:\:{SR}=\mathrm{3}\:\:{TR}=\mathrm{1}\rightarrow\mathrm{9}{ways} \\ $$$$\left.\mathrm{3}\right){FR}=\mathrm{3}\:\:{SR}=\mathrm{2}\:{TR}=\mathrm{1}\rightarrow\mathrm{9}{ways} \\ $$$$\left.\mathrm{4}\right){FR}=\mathrm{2}\:\:{SR}=\mathrm{2}\:\:{TR}=\mathrm{2}\:\:\rightarrow\mathrm{27}{ways} \\ $$$$\left.\mathrm{5}\right){FR}=\:\mathrm{3}\:\:\:{SR}=\mathrm{1}\:\:\:{TR}=\mathrm{2}\:\:\rightarrow\mathrm{9}{ways} \\ $$$$\left.\mathrm{6}\right){FR}=\mathrm{2}\:\:\:{SR}=\mathrm{1}\:\:\:{TR}=\mathrm{3}\:\:\rightarrow\mathrm{9}{ways} \\ $$$$\left.\mathrm{7}\right){FR}=\mathrm{1}\:\:{SR}=\mathrm{3}\:\:\:{TR}=\mathrm{2}\rightarrow\:\mathrm{9}{ways} \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\mathrm{3}{c}_{\mathrm{1}} ×\mathrm{3}{c}_{\mathrm{2}} ×\mathrm{3}{c}_{\mathrm{3}} =\mathrm{9}\:\leftarrow{example}\:{of}\:{calculation}\:{for} \\ $$$${first}\:{case}\:{that}\:{is}\:{FR}=\mathrm{1}\:\:{SR}=\mathrm{2}\:\:{TR}=\mathrm{3} \\ $$$$\left.{others}\:{similar}\:{way}\:{calculated}\right) \\ $$$${so}\:{total}\:\mathrm{9}×\mathrm{6}+\mathrm{27}=\mathrm{81} \\ $$$$ \\ $$

Commented by Necxx last updated on 13/Nov/18

wow.....Thank you so much.

$${wow}.....{Thank}\:{you}\:{so}\:{much}. \\ $$

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