Let →a=→i+→j+→k, be a variable vector such that →r.→i,→r.→j and →r.→k are positive integers. If →r.→a≤12 then the number of values of →r is
A
12C9−1
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B
12C3
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C
12C9
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D
none of these
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Solution
The correct options are B12C3 C12C9
PROBLEMS BASED ON CERTAIN THEOREMS ON COMBINATIONS :
⋅ The number of positive integral solutions of the equation x1+x2+x3+....+xr=n is n−1Cr−1.⋅ The number of non-negative integral solutions of the equationx1+x2+x3+....+xr=n is n+r−1Cr−1.
Let →r=xi+yj+zk ⇒x+y+z≤12 Thus number of positive integral solution is =12−1C3−1+11−1C3−1+...........+4−1C3−1=12C3=12C9,(∵nCr=nCn−r)