2009 AIME II Problems/Problem 6

Problem

Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$ .

Solution

We can use complementary counting. We can choose a five-element subset in ${14\choose 5}$ ways. We will now count those where no two numbers are consecutive. We will show a bijection between this set, and the set of 10-element strings that contain 5 $A$ s and 5 $B$ s, thereby showing that there are ${10\choose 5}$ such sets.

Given a five-element subset $S$ of $\{1,2,\dots,14\}$ in which no two numbers are consecutive, we can start by writing down a string of length 14, in which the $i$ -th character is $A$ if $i\in S$ and $B$ otherwise. Now we got a string with 5 $A$ s and 9 $B$ s. As no two numbers were consecutive, we know that in our string no two $A$ s are consecutive. We can now remove exactly one $B$ from between each pair of $A$ s to get a string with 5 $A$ s and 5 $B$ s. And clearly this is a bijection, as from each string with 5 $A$ s and 5 $B$ s we can reconstruct one original set by reversing the construction.

Hence we have $m = {14\choose 5} - {10\choose 5} = 2002 - 252 = 1750$ , and the answer is $1750 \bmod 1000 = \boxed{750}$ .

Solution 2

Let $A$ be the number of ways in which $5$ distinct numbers can be selected from the set of the first $14$ natural numbers, and let $B$ be the number of ways in which $5$ distinct numbers, no two of which are consecutive, can be selected from the same set. Then $m = A -B$ . Because $A = \dbinom{14}{5}$ , the problem is reduced to finding $B$ . Consider the natural numbers $1 \leq a_1 < a_2 < a_3 < a_4 < a_5 \leq 14$ . If no two of them are consecutive, the numbers $b_1 = a_1, b_2 = a_2 - 1$ , $b_3 = a_3 - 2$ , $b_4 = a_4 - 3$ , and $b_5 = a_5 - 4$ are distinct numbers from the interval $[1, 10]$ . Conversely, if $b_1 < b_2 < b_3 < b_4 < b_5$ are distinct natural numbers from the interval $[1, 10]$ , then $a_1 = b_1$ , $a_2 = b_2 + 1$ , $a_3 = b_3 + 2$ , $a_4 = b_4 + 3$ , and $a_5 = b_5 + 4$ are from the interval $[1, 14]$ , and no two of them are consecutive. Therefore counting $B$ is the same as counting the number of ways of choosing $5$ distinct numbers from the set of the first $10$ natural numbers. Thus $B = \dbinom{10}{5}$ . Hence $m = A - B = \dbinom{14}{5} - \dbinom{10}{5} = 2002 - 252 = 1750$ and the answer is $750$ .

2009 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2009 AIME II Problems/Problem 6

Contents

Problem

Solution

Solution 2

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