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shefali sharma Grade: 12
        

let vector a=i+j+k and let r be a variable vector such that r.i , r.jand r.k are positive integers. if r.a<=12 then what are the no. of values of r?

7 years ago

Answers : (1)

Badiuddin askIITians.ismu Expert
147 Points
										

Dear shefali


 let r =xi +yj +zk




r.i =x ≥1


r.j=y  ≥1


r.k=z  ≥1


and r.a =x+y+z ≤12


  x+y+z ≤12


let u=x-1


   v=y-1


  w=z-1


 then   u+v+w ≤9     where u≥0,v≥0,w≥0


now we have to find the intiger solution of this equation.


let  u+v+w =n   where n varies from 0 to 9


  solution of these equation


 n:0 to 9  ∑n+3-1C3-1


=n:0 to 9  ∑n+2C2


=n:0 to 9  ∑(n+2)(n+1)/2


=n:0 to 9  1/2 ∑(n2+3n+2)


= 1/2 [9(9+1)(18+1)/6   + 3*9(9+1)/2  + 2 *10]


 =1/2[285 +135+20] =220


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Badiuddin

7 years ago
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