text: # Number of positive integral solutions to equation x+y+z+w=10 (x>=0,y>=0,z>=2,w>=2) isPlZZZZZZZZZZ answer soon.
  
  jagdish singh singh
  173 Points
  7 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
  7 years ago
  Sorry I have calculate for non negative Integral solution...................................................
  jagdish singh singh
  173 Points
  7 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
  7 years ago
  But the answer given is 84!!!!!!!!!.How????????????????.Plz provide the answer in a complete manner.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
  
  jagdish singh singh
  173 Points
  7 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
  7 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???????????
explanation: The document contains a math question and several answers to it. The question is about finding the number of positive integral solutions to the equation \(x + y + z + w = 10\) with the constraints \(x \geq 0\), \(y \geq 0\), \(z \geq 2\), and \(w \geq 2\). Multiple users provide different solutions and explanations, with the final correct answer being 84.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Number of positive integral solutions to equation x+y+z+w=10 (x>=0,y>=0,z>=2,w>=2) isPlZZZZZZZZZZ answer soon.
  
  jagdish singh singh
  173 Points
  7 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
  7 years ago
  Sorry I have calculate for non negative Integral solution...................................................
  jagdish singh singh
  173 Points
  7 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
  7 years ago
  But the answer given is 84!!!!!!!!!.How????????????????.Plz provide the answer in a complete manner.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
  
  jagdish singh singh
  173 Points
  7 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
  7 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???????????
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and several answers to it. The question is about finding the number of positive integral solutions to the equation \(x + y + z + w = 10\) with the constraints \(x \geq 0\), \(y \geq 0\), \(z \geq 2\), and \(w \geq 2\). Multiple users provide different solutions and explanations, with the final correct answer being 84.
  
  Label: question-and-answer