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
AB
A
B
BA
6\times 2=12
ST
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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
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