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Number of positive integral solutions to equation x+y+z+w=10 (x>=0,y>=0,z>=2,w>=2) is PlZZZZZZZZZZ answer soon.

Number of positive integral solutions to equation x+y+z+w=10 (x>=0,y>=0,z>=2,w>=2) is
PlZZZZZZZZZZ answer soon.

Grade:12

6 Answers

jagdish singh singh
173 Points
5 years ago
\hspace{-0.70 cm}$Let $z-2=t\geq 0$ and $w-2=u\geq 0$ So eqn.convert into\\\\ $x+y+t+2+u+2=10\Rightarrow x+y+z+u=8\;,$ Where $x,y,z,u\geq0$\\\\Which is equivalent to arrange $8$ stars and $3$ bars in row in that way.\\\\$\underbrace{*}_{x}|\underbrace{-}_{y}|\underbrace{****}_{z}|\underbrace{***}_{u}$, He $x=1\;,y=0,z= 4,u=3$ is one example\\\\ S Total no.of ways of arranging $8$ stars and $3$ bars Row is$ = \binom{8!}{3!\times 5!}$
jagdish singh singh
173 Points
5 years ago
Sorry I have calculate for non negative Integral solution...................................................
jagdish singh singh
173 Points
5 years ago
To Admin when i post my answer it did not take My answer and show a message answer already exists.plz help me. anywhy answer is $\displaystyle \binom{11}{3}$
gandhi1234
10 Points
5 years ago
But the answer given is 84!!!!!!!!!.How????????????????.Plz provide the answer in a complete manner.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
 
jagdish singh singh
173 Points
5 years ago
Actually You are Right Answer is =84. Here is my solution.......................................................\hspace{-.70 cm}$ Let $x+1=x`\geq 1\;,y+1=y`\geq 1$ and $z-1=z`\geq 1\;,u-1=u`\geq 1$\\\\So equation is $x`+y`+z`+u`=10\;,$ Where $x`,y`,z`,u`\geq 0$\\\\Which is equivalent to arange $3$ bars in between $10$ stars.\\\\ Like this way $\underbrace{***}_{x`}|\underbrace{**}_{y`}|\underbrace{****}_{z`}|\underbrace{*}_{u`}$\\\\So Total no. of ways is $=\binom{9}{3} = 84.$
gandhi1234
10 Points
5 years ago
But z>=2 and w>=2 and you take z>=1 and w>=1.How is it possible?My answer is coming out to be 7C3.Is it correct???????????

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