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Question

The position vector of a point P is ¯r=x^i+y^j+z^k where x,y,zϵN and ¯a=^i+^j+^k. If ¯r¯a=12, then number of possible position of P is

A
45
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B
55
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C
35
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D
None of these
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Solution

The correct option is B 55
Let r=xi+yj+zk
Thus, r.a=x+y+z
Hence, x+y+z=12
=>x+y+z=12
This is thus equivalent to distributing 12 identical sweets among 3 boys.
Now, since x, y, z are positive integers, we give them 1 sweet each. Thus, we are left with 123=9
Thus, we need to distribute 9 sweets among 3 boys where a boy getting no sweet is also allowed.
This can be done in9+31C31=11C2=55 ways.
Hence, (b) is correct.

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