Number of shortest ways in which we can reach from the point \( (0,0,0) \) to point \( (3,7,11) ...

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Number of shortest ways in which we can reach from the point \( (0,0,0) \) to point \( (3,7,11) \) in space where the movement is possible only along the \( x \)-axis, \( y \)-axis, and \( z \)-axis or parallel to them and change of axes is permitted only at integral points (an integral point is one which has its coordinate as integer) is
(1) equivalent to number of ways of dividing 21 different objects in three groups of size \( 3,7,11 \)
(2) equivalent to coefficient of \( y^{3} z^{7} \) in the expansion of \( (1+y+z)^{21} \)
(3) equivalent to number of ways of distributing 21 different objects in three boxes of size \( 3,7,11 \)
(4) equivalent to number of ways of arranging 21 objects of which 3 are alike of one kind, 7 are alike of second type, and 11 are alike of third type
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