(Solved): A bug jumps from lattice point to lattice point on a piece of graph paper, one jump per second, ac ...

A bug jumps from lattice point to lattice point on a piece of graph paper, one jump per second, according to the following pattern: From \( (m, n) \), the bug jumps only to \( (m+1, n) \) or \( (m, n+1) \), each equally likely. Find all the places the bug could be, two seconds after it leaves the origin \( (0,0) \). Are all these places equally likely? (Continuation) Where could the bug be, three seconds after it leaves the origin? Are all these places equally likely? (Continuation) Expand \( (r+u)^{2} \) and \( (r+u)^{3} \), looking for connections with the jumping bug. In algebra, it is customary to collect terms like \( r r u \), rur, and \( u r r \) into a single term \( 3 r^{2} u \). Is there any reason to distinguish these terms in this example, however?