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?

Short answer

Therefore, there are\(50,450,000\)ways to travel in xyz space from the origin to \((0,0,0,0)\;to\;(4,3,5,4)\)point by taking the described steps.

Step-by-step solution

Step 1: Assumptions and substitution

To move from\((0,0,0,0)\;to\;(4,3,5,4)\) we will have to travel four times in the positive x-direction, three times in the positive y-direction, and five times in the position z-direction and four times in the positive w-direction

\(\begin{array}{l}n = 4 + 3 + 5 + 4 = 16\\k = 3\end{array}\)

\(\begin{array}{l}{n_1} = 4\\{n_2} = 3\\{n_3} = 5\\{n_4} = 4\end{array}\)

Step 2: Further simplification

Distributing n distinguishable objects into k distinguishable boxes such that \({n_i}\) objects are placed in box i (\(i = 1,2,3,4,5\)) can be done in \(\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}} = \frac{{16!}}{{4!3!5!4!}} = 50,450,000\)

Therefore, there are\(50,450,000\)ways to travel in xyz space from the origin to \((0,0,0,0)\;to\;(4,3,5,4)\)point by taking the described steps.