The number of ways to reach the point $Q (4, 5, 3)$ from point $P (1, 2, 1)$ in the space where movement is only allowed parallel to the three axes and change of direction is allowed only at the integral points is

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Solution

To reach point $Q$ from $P$ , required steps along $x, y$ and $z$ axis are
$| x_{2} - x_{1} | = 4 - 1 = 3 | y_{2} - y_{1} | = 5 - 2 = 3$
and
$| z_{2} - z_{1} | = 3 - 1 = 2$ respectively.

Total number of steps $= 8$
Required number of ways to reach point $Q$ from $P$ is,
$= \frac{8!}{3! \times 3! \times 2!} = 560$

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The number of ways to reach the point Q(4,5,3) from point P(1,2,1) in the space where movement is only allowed parallel to the three axes and change of direction is allowed only at the integral points is

To reach point Q from P, required steps along x,y and z axis are |x2−x1|=4−1=3|y2−y1|=5−2=3 and |z2−z1|=3−1=2 respectively. Total number of steps =8 Required number of ways to reach point Q from P is, =8!3!×3!×2!=560

The number of ways to reach the point $Q (4, 5, 3)$ from point $P (1, 2, 1)$ in the space where movement is only allowed parallel to the three axes and change of direction is allowed only at the integral points is