Difference between revisions of "2012 AMC 10A Problems/Problem 23"

Revision as of 17:34, 10 August 2021

The following problem is from both the 2012 AMC 12A #19 and 2012 AMC 10A #23, so both problems redirect to this page.

Problem

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?

$\text{(A)}\ 60\qquad\text{(B)}\ 170\qquad\text{(C)}\ 290\qquad\text{(D)}\ 320\qquad\text{(E)}\ 660$

Solution

Note that if $n$ is the number of friends each person has, then $n$ can be any integer from $1$ to $4$ , inclusive.

Also note that one person can only have at most 5 friends.

Also note that the cases of $n=1$ and $n=4$ are the same, since a map showing a solution for $n=1$ can correspond one-to-one with a map of a solution for $n=4$ by simply making every pair of friends non-friends and vice versa. The same can be said of configurations with $n=2$ when compared to configurations of $n=3$ . Thus, we have two cases to examine, $n=1$ and $n=2$ , and we count each of these combinations twice.

For $n=1$ , if everyone has exactly one friend, that means there must be $3$ pairs of friends, with no other interconnections. The first person has $5$ choices for a friend. There are $4$ people left. The next person has $3$ choices for a friend. There are two people left, and these remaining two must be friends. Thus, there are $15$ configurations with $n=1$ .

For $n=2$ , there are two possibilities. The group of $6$ can be split into two groups of $3$ , with each group creating a friendship triangle. The first person has $\binom{5}{2} = 10$ ways to pick two friends from the other five, while the other three are forced together. Thus, there are $10$ triangular configurations.

However, the group can also form a friendship hexagon, with each person sitting on a vertex, and each side representing the two friends that person has. The first person may be seated anywhere on the hexagon without loss of generality. This person has $\binom{5}{2} = 10$ choices for the two friends on the adjoining vertices. Each of the three remaining people can be seated "across" from one of the original three people, forming a different configuration. Thus, there are $10 \cdot 3! = 60$ hexagonal configurations, and in total $70$ configurations for $n=2$ .

As stated before, $n=3$ has $70$ configurations, and $n=4$ has $15$ configurations. This gives a total of $(70 + 15)\cdot 2 = 170$ configurations, which is option $\boxed{\textbf{(B)}\ 170}$ .

Note

We can also calculate the triangular configurations by applying $\frac{\binom{6}{3}}{2} = \frac{20}{2}=10$ (Because choosing $A$ , $B$ and $C$ is the same as choosing $D$ , $E$ and $F$ .

For the hexagonal configurations, we know that the total amount of combinations is $6!=720$ . However, we must correct for our overcounting because of rotation and reflection. We have that there are ${720 \over 6 \cdot 2} = 60$ because there $6$ rotations of the hexagon and 2 reflections (clockwise and counterclockwise) for each valid way.

Note 2

We can also calculate the hexagonal configurations by placing each person at a random vertex (6!) and dividing for rotations (by 6) and reflections (by 2):

Lamsheeper

Another Way

For the hexagonal configuration, it is essentially congruent to counting the number of paths that start and end with A. From $A$ , we have $5$ options, then the next one has $4$ , and so on, so we have: $5\cdot 4 \hdots 1 \cdot 1$ , where the final $1$ is because the final point needs to reconnect to $A$ . Thus, there are $5! = 120$ ways. Dividng by $2$ since we overcounted by accounting for both directions, we have $120/2 = \fbox{60}$

-bigbrain123

Video Solution by Richard Rusczyk

https://www.youtube.com/watch?v=92X9ePsgPRU

@@ Line 38: / Line 38: @@
 Lamsheeper
+=== Another Way ===
+For the hexagonal configuration, it is essentially congruent to counting the number of paths that start and end with A. From <math>A</math>, we have <math>5</math> options, then the next one has <math>4</math>, and so on, so we have: <math>5\cdot 4 \hdots 1 \cdot 1</math>, where the final <math>1</math> is because the final point needs to reconnect to <math>A</math>. Thus, there are <math>5! = 120</math> ways. Dividng by <math>2</math> since we overcounted by accounting for both directions, we have <math>120/2 = \fbox{60}</math>
+-bigbrain123
 ==Video Solution by Richard Rusczyk==

2012 AMC 10A (Problems • Answer Key • Resources)
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