Question: How many different paths are there through the pinball machine?

To answer this question we will count the number of different paths at each row of pins as the pinball progresses through the machine and list them:

At the first pin there are 2 possible deflections, L and R. Thus to this point:

the number of paths is:   2
the list of paths is:         {L R}

At the second row of pins there is again a deflection, either L or R. Since what happens at the second pin is independent of what happened at the first pin we can multiply this number of possiblities (2) by the previous number of possibilities (2) to get the number of possible paths to this point (this is called the multiplication rule for independent events):

the number of paths is:   2 x 2 = 22 = 4
the list of paths is:         {LL   LR   RL   RR}

At the third row there is again a deflection, L or R, which is independent of what happened previously so we again multiply this number of possiblities (2) by the previous number of possibilities (4):

the number of paths is:   2 x 2 x 2 = 23 = 8
the list of paths is:         {LLL   LLR   LRL   LRR   RLL   RLR   RRL   RRR}
(The easiest way to get this list is to list the numbers 0 to 7 using binary numbers: 000,   001,   010,   011,   100,   101,   110,   111. Then replace each 0 by L and each 1 by R.)


At the fourth row the 2 possible deflections, L and R, are again multiplied by the previous number of possibilities (8), and we get:

the number of paths is:   2 x 2 x 2 x 2 = 24 = 16
the list of paths is:
LLLL   LLLR   LLRL   LLRR
LRLL   LRLR   LRRL   LRRR
RLLL   RLLR   RLRL   RLRR
RRLL   RRLR   RRRL   RRRR
Answer: There are 16 different paths through the machine.

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