2012 AIME I Problems/Problem 3
Problem 3
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person.
Solution 1
Call a beef meal a chicken meal
and a fish meal
Now say the nine people order meals
respectively and say that the person who receives the correct meal is the first person. We will solve for this case and then multiply by
to account for the
different ways in which the person to receive the correct meal could be picked. Note, this implies that the dishes are indistinguishable, though the people aren't. For example, two people who order chicken are separate, though if they receive fish, there is only 1 way to order them.
The problem we must solve is to distribute meals to orders
with no matches. The two people who ordered
's can either both get
's, both get
's, or get one
and one
We proceed with casework.
- If the two
people both get
's, then the three
meals left to distribute must all go to the
people. The
people then get
in some order, which gives three possibilities. The indistinguishability is easier to see here, as we distribute the
meals to the
people, and there is only 1 way to order this, as all three meals are the same.
- If the two
people both get
's, the situation is identical to the above and three possibilities arise.
- If the two
people get
in some order, then the
people must get
and the
people must get
This gives
possibilities.
Summing across the cases we see there are possibilities, so the answer is
Solution 2
We only need to figure out the number of ways to order the string , where exactly one
is in the first three positions, one
is in the
to
positions, and one
is in the last three positions. There are
ways to place the first
meals. Then for the other two people, there are
ways to serve their meals. Thus, there are
ways to serve their meals.
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2012aimei/331
~ dolphin7
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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