2020 AMC 8 Problems/Problem 10
Zara has a collection of marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
Solution
Let the Aggie, Bumblebee, Steelie, and Tiger, be referred to by and
, respectively. If the ignore the constraint that
and
cannot be next to each other, we get a total of
ways to arrange the 4 marbles. We now simply have to subtract out the number of ways that
and
can be next to each other. If we place
and
next to each other in that order, then there are three places that we can place them, namely in the first two slots, in the second two slots, or in the last two slots (i.e. $ST\fbox\fbox$ (Error compiling LaTeX. Unknown error_msg)). However, we could also have placed
and
in the opposite order (i.e. $TS\fbox\fbox$ (Error compiling LaTeX. Unknown error_msg)). Thus there are 6 ways of placing
and
A
B
A
B
A
B
B
A
6\times 2=12
S
T
24-12=12
\implies\boxed{\textbf{(C) }12}$.
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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