Probability Question...seems easy but can't figure it out

[TearinMyHairOut]

Can't seem to figure this one out for the life of me...well I'm pretty sure I know the answer (19), but can't figure out how to actually get there, any help would be greatly appreciated:

A runner runs 2 races on a ten lane track. He is given a lane assignment consisting of one lane for the first race and one lane for the second race. How many lane assignments are there in which he gets lane one for at least one race? (A lane assignment consists of a pair of numbers e.g. (3,4) meaning 3 for race 1 and 4 for race 2.)

 

[tkhunny]

With such a small number, why not just count them?

(1,1)
(1,2)
(1,3)
etc.
(1,10)
(2,1)
(3,1)
etc
(10,1)

How many are there?

Don't count (1,1) twice.

 

[TearinMyHairOut]

i would (and did)...but my professor will look at that problem and stamp a big ol' zero on it...is there a way to complete this problem without just simply counting the answers?

 

[mmm4444bot]

… my professor will look at that [answer] and stamp a big ['ol] zero on it …

How cruel.

I would eliminate all of the numbers from the "counted" answer and repackage it (i.e., disguise it) in razzle-dazzle, instead.

Something along the lines of the following, perhaps?

Clearly, in all such ordered pairs, there are only three possible cases: (1) the runner is assigned to lane one for the first race only; (2) the runner is assigned to lane one for the second race only; (3) the runner is assigned to lane one for both races. In each of the cases (1) and (2), since the runner occupies lane one, there are only nine lanes remaining from which to choose for the other assignment. Therefore, cases (1) and (2) account for eighteen possible ordered pairs in which either the first number is one or the second number is one and both numbers are not one. Adding the single ordered pair from case (3) gives nineteen possible lane assignments that satisfy the given senario.