Kenneth Rosen Edition 7 Exercise 6.5 Question 40 (Page No. 433)

Question

How many ways are there to travel in $xyzw$ space from the origin $(0, 0, 0, 0)$ to the point $(4, 3, 5, 4)$ by taking steps one unit in the positive $x,$ positive $y,$ positive $z,$ or positive $w$ direction?

Answer 1

$\binom{16}{4}\cdot \binom{16-4}{3}\cdot \binom{12-3}5{}$=50450400

The net displcement will be 4 in x, 3 in y, 5 in z and 4 in w direction.

The number of movements in total in all direction=4+3+5+4=16

By choosing any 4 movements in x dirn, 3 in y, 5 in z and 4 in w