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Stage 4 · Combinatorics

10 problems · AMC 12 late, AIME early · mathsolympiadprep.com

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1. Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $400$ feet or less to the new gate be a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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2. Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\,$ or $10''\,$ or $19''\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?

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3. Some students participate in three competitions, the first place gets 5 points, the second place gets 3 points, the third place gets 1 point, and no points for not placing. Then, how many points must a student get to definitely have more points than another student?

   1. A9
   2. B10
   3. C11
   4. D13
   5. E15

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4. As shown in Figure 7, the shaded part is composed of three small squares on the grid paper, and such a pattern is called an "L-shape". Therefore, on a grid paper composed of $4 \times 5$ small squares, the number of different L-shape patterns that can be drawn is:

   1. A16
   2. B32
   3. C48
   4. D64

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5. Three Shepherds. When the Crackhams approached a large city, they had to stop because a flock of sheep was moving along the road, followed by a herd of bulls, and then the shepherds were driving a herd of horses. The Crackhams realized that it was market day in the city today. George, taking advantage of the opportunity, came up with the following puzzle.

   Three shepherds driving their herds met on a highway. Jack said to Jim:

   - If I give you 6 pigs for one horse, then your herd will have twice as many heads as mine.

   And Dan remarked to Jack:

   - If I give you 14 sheep for one horse, then your herd will have three times as many heads as mine.

   Jim, in turn, said to Dan:

   - And if I give you 4 cows for one horse, then your herd will be six times larger than mine.

   The deals did not go through, but could you still tell how many heads of livestock there were in the three herds.

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6. A magazine prints six photos, which are of three

   famous people and their baby photos, but the three baby photos are not labeled as to which three people they belong to, and readers have to choose themselves. Assuming each photo is equally likely. Then the probability that a reader randomly selects three photos, making the baby photos correspond to the names of the three celebrities is:

   1. A$\frac{1}{9}$
   2. B$\frac{1}{6}$
   3. C$\frac{1}{4}$
   4. D$\frac{1}{3}$
   5. E$\frac{1}{2}$

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7. In the annual football league of a certain middle school, it is stipulated that each team must play one match against every other team. If in the 2012 season all teams played a total of 21 matches. Then the number of participating teams is:

   1. A6
   2. B7
   3. C8
   4. D9
   5. E10

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8. Which two weights need to be swapped to balance the scale? (The unit of all weights is grams)

   1. A1 and 4
   2. B9 and 2
   3. C9 and 4
   4. D7 and 3
   5. E7 and 4

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9. A7. On a bicycle lock, we can set a 3-digit code. For each of the three positions, we can choose one of the digits from 0 to 9. What is the maximum number of different codes we can set?

   1. A30
   2. B100
   3. C300
   4. D$9^{3}$
   5. E1000

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10. In how many ways can three people be selected for three identical positions from ten candidates?

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Answer key — Stage 4 · Combinatorics

Worked solutions for every problem are on the site, one page per problem.

1. $52$ open
2. $465$ open
3. $D$ open
4. $C$ open
5. $39$ open
6. $\frac{1}{6}$ open
7. $B$ open
8. $D$ open
9. $1000$ open
10. $120$ open

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