Difference between revisions of "2012 AIME II Problems/Problem 14"
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f(5): There are <math>\dbinom{4}{2}=6</math> possible ways for person A to shake hands with two others (WLOG assume they are B and C). Then, B and C must also shake hands with the remaining two people (2 ways), and those last 2 people must shake hands with each other (1 way). | f(5): There are <math>\dbinom{4}{2}=6</math> possible ways for person A to shake hands with two others (WLOG assume they are B and C). Then, B and C must also shake hands with the remaining two people (2 ways), and those last 2 people must shake hands with each other (1 way). | ||
− | Therefore, f(5)=<math>\dbinom{4}{2}*2*1=12 | + | Therefore, f(5)=<math>\dbinom{4}{2}*2*1=12</math> |
− | f(6): There are < | + | f(6): There are <math>\dbinom{5}{2}=10</math> possible ways for person A to shake hands with two others (WLOG assume they are B and C). However, B and C shakes hands with each other, then there are f(3) ways for the rest of the people to shake hands. If B and C shake hands with two others out of the three remaining people (3 * 2 ways), and those 2 people must shake hands with the last person (1 way). |
− | Therefore, f(5)=< | + | Therefore, f(5)=<math>\dbinom{5}{2}*(f(3)+3*2*1)=70 |
Now we have enough information to find f(9) | Now we have enough information to find f(9) | ||
− | f(9): There are <math>\dbinom{8}{2}=28< | + | f(9): There are </math>\dbinom{8}{2}=28<math> possible ways for person A to shake hands with two others (WLOG assume they are B and C). |
Starting situation: | Starting situation: | ||
A: B C B: A C: A D: E: F: G: H: I: | A: B C B: A C: A D: E: F: G: H: I: | ||
Line 64: | Line 64: | ||
There are 2*1 ways for F and G to then shake hands with H and I, and H and I will shake hands with each other. | There are 2*1 ways for F and G to then shake hands with H and I, and H and I will shake hands with each other. | ||
− | We have 28(f(6)+6*f(5)+6*5*(f(4)+4*f(3)+4*3*(2*1)))=30016, which gives us the answer of \boxed{016} | + | We have 28(f(6)+6*f(5)+6*5*(f(4)+4*f(3)+4*3*(2*1)))=30016, which gives us the answer of \boxed{016}</math>. |
==Video Solution== | ==Video Solution== |
Revision as of 19:54, 14 December 2022
Problem 14
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when
is divided by
.
Solution 1
Given that each person shakes hands with two people, we can view all of these through graph theory as 'rings'. This will split it into four cases: Three rings of three, one ring of three and one ring of six, one ring of four and one ring of five, and one ring of nine. (All other cases that sum to nine won't work, since they have at least one 'ring' of two or fewer points, which doesn't satisfy the handshaking conditions of the problem.)
Case 1: To create our groups of three, there are . In general, the number of ways we can arrange people within the rings to count properly is
, since there are
ways to arrange items in the circle, and then we don't want to want to consider reflections as separate entities. Thus, each of the three cases has
arrangements. Therefore, for this case, there are
Case 2: For three and six, there are sets for the rings. For organization within the ring, as before, there is only one way to arrange the ring of three. For six, there is
. This means there are
arrangements.
Case 3: For four and five, there are sets for the rings. Within the five, there are
, and within the four there are
arrangements. This means the total is
.
Case 4: For the nine case, there is arrangement for the ring. Within it, there are
arrangements.
Summing the cases, we have .
Solution 2
Let f(N) be the number of ways a group of N people can shake hands with exactly two of the other people from the group, where N \ge 3.
We can easily find that f(3)=1.
Continuing on, we will label the people as A,B,C,... corresponding with N.
f(4): There are possible ways for person A to shake hands with two others. The situation is illustrated below (WLOG assume A shakes hands with B and C), with the letters after the semicolon representing the people who shake the person before the semicolon. Note that a person cannot shake more than two hands.
A: B C B: A C: A D:
The only possible ending of this situation is shown below A: B C B: A D C: A D D: B C
Therefore, we can conclude that f(4)=3
f(5): There are possible ways for person A to shake hands with two others (WLOG assume they are B and C). Then, B and C must also shake hands with the remaining two people (2 ways), and those last 2 people must shake hands with each other (1 way).
Therefore, f(5)=
f(6): There are possible ways for person A to shake hands with two others (WLOG assume they are B and C). However, B and C shakes hands with each other, then there are f(3) ways for the rest of the people to shake hands. If B and C shake hands with two others out of the three remaining people (3 * 2 ways), and those 2 people must shake hands with the last person (1 way).
Therefore, f(5)=$\dbinom{5}{2}*(f(3)+3*2*1)=70
Now we have enough information to find f(9) f(9): There are$ (Error compiling LaTeX. Unknown error_msg)\dbinom{8}{2}=28$possible ways for person A to shake hands with two others (WLOG assume they are B and C). Starting situation: A: B C B: A C: A D: E: F: G: H: I:
'''Case 1:''' B and C shakes hands with each other A: B C B: A C C: A B D: E: F: G: H: I: There are f(6) ways for D,E,F,G,H, and I to shake hands
'''Case 2:''' B and C shake hands with one of the others out of the 6 remaining people (6 ways) A: B C B: A D C: A D D: B C E: F: G: H: I: There are f(5) ways for E,F,G,H, and I to shake hands
'''Case 3:''' If B and C shake hands with two different people (there are 6*5 ways to choose the people but WLOG assume they are D and E) Calculations of Case 3 are in its subcases. A: B C B: A D C: A E D: B E: C F: G: H: I:
'''Subcase 3.1:''' D and E shake hands with each other A: B C B: A D C: A E D: B E E: C D F: G: H: I: There are f(4) ways for F,G,H, and I to shake hands
'''Subcase 3.2:''' D and E each shake hands with one other person (there are 4 ways to choose the person, WLOG let that be F) A: B C B: A D C: A E D: B F E: C F F: D E G: H: I: There are f(3) ways for G,H, and I to shake hands
'''Subcase 3.3:''' D and E each shake hands with two different people (there are 4*3 ways, WLOG let that be F and G)
A: B C B: A D C: A E D: B F E: C G F: D G: E H: I:
There are 2*1 ways for F and G to then shake hands with H and I, and H and I will shake hands with each other.
We have 28(f(6)+6*f(5)+6*5*(f(4)+4*f(3)+4*3*(2*1)))=30016, which gives us the answer of \boxed{016}$ (Error compiling LaTeX. Unknown error_msg).
Video Solution
Very Neat solution: https://www.youtube.com/watch?v=lG8N9RuI-8o
Similar Problems
2006 HMMT feb. combo #9.
See Also
2012 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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