Difference between revisions of "2025 AIME I Problems/Problem 7"

Revision as of 20:52, 13 February 2025

Problem

The twelve letters $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$ , $CJ$ , $DG$ , $EK$ , $FL$ , $HI$ . The probability that the last word listed contains $G$ is $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution 1

Note that order does not matter here. This is because any permutation of the $6$ pairs will automatically get ordered in alphabetical order. The same is true for within each of the pairs. In other words, AB CH DI EJ FK GL should be counted equally as HC AB DI EJ FK GL.

We construct two cases: $G$ is the first letter of the last word and $G$ is the second letter of the last word.

Our first case is when $G$ is the first letter of the last word. Then the second letter of the last word must be one of $H, I, J, K, L$ . Call that set of $5$ letters $\Omega$ . There are $5$ ways to choose the second letter from $\Omega$ . The other $4$ letters of $\Omega$ must be used in the other $5$ words.

For the other 5 words, each of their first letters must be before $G$ in the alphabet. Otherwise, the word with $G$ will not be the last. There are $6$ letters before $G$ : $A,B,C,D,E,F$ . Call that set of $6$ letters $\Sigma$ . Exactly one of the words must have two letters from $\Sigma$ . The other 4 will have their first letter from $\Sigma$ and the second letter from $\Omega$ . There are $4!$ ways to determine the possible pairings of letters from $\Sigma$ and $\Omega$ , respectively.

Therefore, this case has $5 \cdot {6\choose{2}} \cdot 4! = 5 \cdot 15 \cdot 24 = 1800$ orderings.

The second case is when $G$ is the second letter of the last word. You can see that the first letter of that word must be $F$ . Otherwise, that word cannot be the last word. The other $5$ words must start with $A$ , $B$ , $C$ , $D$ , and $E$ . The second letter of each of those words will come from $\Omega$ . There will be $5!$ ways to distribute the elements of $\Omega$ to one of $A, B, C, D, E$ . There are therefore $5! = 120$ orderings in the case.

In total, there are $1800+120 = 1920$ orderings. However, we want the probability. The number of ways to put the $12$ letters into pairs is $11 \cdot 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1$ . This is true because we can say this: Start with $A$ . It has $11$ options for who it will partner with. There are now $10$ letters left. Pick one of those letters. It has $9$ options for who it will partner with. There are now $8$ letters left. Continue until there are only $2$ letters left, and there is only $1$ option for that last word. Therefore, there will be $11 \cdot 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1$ options.

The probability is therefore $\frac{1920}{11 \cdot 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1} = \frac{128}{693}$ . The requested answer is $128 + 693 = \boxed{821}$ .

~lprado

Solution 2: Same but quicker

Splitting up into 2 cases: G is the first letter or the second letter of the last word.

Case 1: G in first letter Notice that A must take the first letter of first word, one of the letters B-F needs to be the second letter of a word and the rest being the first letter of a word. The combinations will be 1 + 2 + 3 + 4 + 5 = 15. After the first 7 letters has been decided then the last 5 will just fill by 5!. This case will have 15 * 5! outcomes.

Case 2: G in last letter Notice that A-G has been arranged by A? B? C? D? E? FG, where the ? is undecided. We have another 5! to fill out the possible outcomes.

In total, there are 16 * 5!. The total case will be 11 * 9 * 7 * 5 * 3 * 1 (Consider A must be in the first letter of first word, then you have 11 choices, then you must take the next letter in alphabetical order as mandatory, then you have a free choice of 9 and so on).

Answer: = 16 * 5 * 4 * 3 * 2 * 1 / 11 * 9 * 7 * 5 * 3 * 1 = 16 * 4 * 2 / 11 * 9 * 7 = 128 / 693 Therefore it gives us the answer of 128 + 693 = 821.

(I will really appreciate if anyone could latex format it better) ~Mitsuihisashi14

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Difference between revisions of "2025 AIME I Problems/Problem 7"

Revision as of 20:52, 13 February 2025

Contents

Problem

Solution 1

Solution 2: Same but quicker

See also

@@ Line 32: / Line 32: @@
 Notice that A-G has been arranged by A? B? C? D? E? FG, where the ? is undecided. We have another 5! to fill out the possible outcomes.
-In total, there are 16 * 5!. The total case will be 11 * 9 * 7 * 5 * 3 * 1.
+In total, there are 16 * 5!. The total case will be 11 * 9 * 7 * 5 * 3 * 1 (Consider A must be in the first letter of first word, then you have 11 choices, then you must take the next letter in alphabetical order as mandatory, then you have a free choice of 9 and so on).
-Answer = 16 * 5 * 4 * 3 * 2 * 1 / 11 * 9 * 7 * 5 * 3 * 1
+Answer:
-       = 16 * 4 * 2 / 11 * 9 * 7
+= 16 * 5 * 4 * 3 * 2 * 1 / 11 * 9 * 7 * 5 * 3 * 1
-       = 128 / 693
+= 16 * 4 * 2 / 11 * 9 * 7
+= 128 / 693
 Therefore it gives us the answer of 128 + 693 = 821.
 (I will really appreciate if anyone could latex format it better)
 ~Mitsuihisashi14
 ==See also==