Bowling Pins

After rolling the first ball of a frame in a game of 10-pin bowling, how many different pin configurations can remain (assuming all configurations are physically possible)?
Source: NCTM Mathematics Teacher, September 2006

Solution
A pin configuration is made up of ten pins each of which can be up or down. The number of possible configurations equals 2^{10}=1024.

Answer: 1024

Alternative solution
Let n represent the number of pins left standing after a roll. In how many ways can they be left standing?
\binom{10}{0}=1 way to leave zero pin (bowl a strike)
\binom{10}{1}=10 ways to leave a single pin
\binom{10}{2}=45 ways to leave two pins
\cdots
\binom{10}{10}=1 way to leave all ten pins (gutter ball)
Total number of possible ways
\binom{10}{0}+\binom{10}{1}+\binom{10}{2}+\binom{10}{3}+\binom{10}{4}+\binom{10}{5}+\binom{10}{6}+\binom{10}{7}+\binom{10}{8}+\binom{10}{9}+\binom{10}{10}=1+10+45+120+210+252+210+120+45+10+1=1024

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About mvtrinh

Retired high school math teacher.
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