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| − | ==Problem==
| + | #redirect [[2022 AMC 10A Problems/Problem 22]] |
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| − | Suppose that 13 cards numbered <math>1, 2, 3, \cdots, 13</math> are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the <math>13!</math> possible orderings of the cards will the <math>13</math> cards be picked up in exactly two passes?
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| − | <asy>
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| − | size(11cm);
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| − | draw((0,0)--(2,0)--(2,3)--(0,3)--cycle);
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| − | label("7", (1,1.5));
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| − | draw((3,0)--(5,0)--(5,3)--(3,3)--cycle);
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| − | label("11", (4,1.5));
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| − | draw((6,0)--(8,0)--(8,3)--(6,3)--cycle);
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| − | label("8", (7,1.5));
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| − | draw((9,0)--(11,0)--(11,3)--(9,3)--cycle);
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| − | label("6", (10,1.5));
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| − | draw((12,0)--(14,0)--(14,3)--(12,3)--cycle);
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| − | label("4", (13,1.5));
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| − | draw((15,0)--(17,0)--(17,3)--(15,3)--cycle);
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| − | label("5", (16,1.5));
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| − | draw((18,0)--(20,0)--(20,3)--(18,3)--cycle);
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| − | label("9", (19,1.5));
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| − | draw((21,0)--(23,0)--(23,3)--(21,3)--cycle);
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| − | label("12", (22,1.5));
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| − | draw((24,0)--(26,0)--(26,3)--(24,3)--cycle);
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| − | label("1", (25,1.5));
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| − | draw((27,0)--(29,0)--(29,3)--(27,3)--cycle);
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| − | label("13", (28,1.5));
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| − | draw((30,0)--(32,0)--(32,3)--(30,3)--cycle);
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| − | label("10", (31,1.5));
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| − | draw((33,0)--(35,0)--(35,3)--(33,3)--cycle);
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| − | label("2", (34,1.5));
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| − | draw((36,0)--(38,0)--(38,3)--(36,3)--cycle);
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| − | label("3", (37,1.5));
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| − | </asy>
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| − | <math>\textbf{(A) } 4082 \qquad \textbf{(B) } 4095 \qquad \textbf{(C) } 4096 \qquad \textbf{(D) } 8178 \qquad \textbf{(E) } 8191</math>
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| − | [[2022 AMC 10A Problems/Problem 22|Solution]] | |
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| − | ==Solution 1==
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| − | Since the <math>13</math> cards are picked up in two passes, the first pass must pick up the first <math>n</math> cards and the second pass must pick up the remaining cards <math>m</math> through <math>13</math>.
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| − | Also note that if <math>m</math>, which is the card that is numbered one more than <math>n</math>, is placed before <math>n</math>, then <math>m</math> will not be picked up on the first pass since cards are picked up in order. Therefore we desire <math>m</math> to be placed before <math>n</math> to create a second pass, and that after the first pass, the numbers <math>m</math> through <math>13</math> are lined up in order from least to greatest.
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| − | To construct this, <math>n</math> cannot go in the <math>n</math>th position because all cards <math>1</math> to <math>n-1</math> will have to precede it and there will be no room for <math>m</math>. Therefore <math>n</math> must be in slots <math>n+1</math> to <math>13</math>.
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| − | Let's do casework on which slot <math>n</math> goes into to get a general idea for how the problem works.
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| − | <math>\textbf{Case 1:}</math> With <math>n</math> in spot <math>n+1</math>, there are <math>n</math> available slots before <math>n</math>, and there are <math>n-1</math> cards preceding <math>n</math>. Therefore the number of ways to reserve these slots for the <math>n-1</math> cards is <math>\binom{n}{n-1}</math>. Then there is only <math>1</math> way to order these cards (since we want them in increasing order). Then card <math>m</math> goes into whatever slot is remaining, and the <math>13-m</math> cards are ordered in increasing order after slot <math>n+1</math>, giving only <math>1</math> way. Therefore in this case there are <math>\binom{n}{n-1}</math> possibilities.
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| − | <math>\textbf{Case 2:}</math> With <math>n</math> in spot <math>n+2</math>, there are <math>n+1</math> available slots before <math>n</math>, and there are <math>n-1</math> cards preceding <math>n</math>. Therefore the number of ways to reserve slots for these cards are <math>\binom{n+1}{n-1}</math>. Then there is one way to order these cards. Then cards <math>m</math> and <math>m+1</math> must go in the remaining two slots, and there is only one way to order them since they must be in increasing order. Finally, cards <math>m+2</math> to <math>13</math> will be ordered in increasing order after slot <math>n+1</math>, which yields <math>1</math> way. Therefore, this case has <math>\binom{n+1}{n-1}</math> possibilities.
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| − | I think we can see a general pattern now. With <math>n</math> in slot <math>x</math>, there are <math>x-1</math> slots to distribute to the previous <math>n-1</math> cards, which can be done in <math>\binom{x-1}{n-1}</math> ways. Then the remaining cards fill in in just <math>1</math> way. Since the cases of <math>n</math> start in slot <math>n+1</math> and end in slot <math>13</math>, this sum amounts to:
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| − | <cmath>\binom{n}{n-1}+\binom{n+1}{n-1}+\binom{n+2}{n-1} + \cdots + \binom{12}{n-1}</cmath> for any <math>n</math>.
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| − | Hmmm... where have we seen this before?
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| − | We use wishful thinking to add a term of <math>\binom{n-1}{n-1}</math>:
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| − | <cmath>\binom{n-1}{n-1}+\binom{n}{n-1}+\binom{n+1}{n-1}+\binom{n+2}{n-1} + \cdots + \binom{12}{n-1}</cmath>
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| − | This is just the hockey stick identity! Applying it, this expression is equal to <math>\binom{13}{n}</math>. However, we added an extra term, so subtracting it off, the total number of ways to order the <math>13</math> cards for any <math>n</math> is <cmath>\binom{13}{n}-1</cmath>
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| − | Finally, to calculate the total for all <math>n</math>, we sum from <math>n=0</math> to <math>13</math>. This yields us:
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| − | <cmath>\sum_{n=0}^{13} \binom{13}{n}-1 \implies \sum_{n=0}^{13} \binom{13}{n} - \sum_{n=0}^{13} 1</cmath>
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| − | <cmath>\implies 2^{13} - 14 = 8192 - 14 = 8178 = \boxed{D}</cmath>
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| − | ~KingRavi
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| − | ==Solution 2==
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| − | To solve this problem, we can use recursion on <math>n</math>. Let <math>A_n</math> be the number of arrangements for <math>n</math> numbers. Now, let's look at how these arrangements are formed by case work on the first number <math>a_1</math>.
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| − | If <math>a_1 = 1</math>, the remaining <math>n-1</math> numbers from <math>2</math> to <math>n</math> are arranged in the same way just like number 1 to <math>n-1</math> in the case of <math>n-1</math> numbers. So there are <math>A_{n-1}</math> arrangements.
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| − | If <math>a_1 = 2</math>, then we need to choose 1 position from position 2 to <math>n-1</math> to put 1, and all remaining numbers must be arranged in increasing order, so there are <math>\binom{n-1}{1}</math> such arrangements.
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| − | If <math>a_1 = k</math>, then we need to choose <math>k-1</math> positions from position 2 to <math>n-1</math> to put <math>1, 2,\cdots k-1</math>, and all remaining numbers must be arranged in increasing order, so there are <math>\binom{n-1}{k-1}</math> such arrangements.
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| − | So we can write
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| − | <cmath>
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| − | A_n = A_{n-1} + \binom{n-1}{1} + \binom{n-1}{2} + \cdots + \binom{n-1}{n-1}
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| − | </cmath>
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| − | which can be simplified to
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| − | <cmath>
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| − | A_n = A_{n-1} + 2^{n-1} - 1
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| − | </cmath>
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| − | We can solve this recursive sequence by summing up <math>n-1</math> lines of the recursive formula
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| − | <cmath>A_n - A_{n-1} = 2^{n-1} - 1</cmath>
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| − | <cmath>A_{n-1} - A_{n-2} = 2^{n-2} - 1</cmath>
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| − | <cmath>\cdots</cmath>
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| − | <cmath>A_2 - A_{1} = 2^{1} - 1</cmath>
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| − | to get
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| − | <cmath>
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| − | A_n - A_1 = \sum_{k=1}^{n-1} (2^k - 1) = 2^n - 2 - (n-1) = 2^n - n - 1
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| − | </cmath>
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| − | since <math>A_1=0</math>, we have
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| − | <cmath>
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| − | A_n = 2^n - n - 1
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| − | </cmath>
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| − | and <math>A_{13} = 2^{13} - 14 = \boxed{8178}</math>.
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| − | So the answer is <math>\boxed{D}</math>
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| − | -- Dan Li
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| − | ==Solution 3==
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| − | Let the cards picked up in the first pass be <math>1, 2, \ldots, n</math>, where <math>1 \leq n \leq 12</math>. Then there are <math>\binom{13}{n} -1</math> ways the cards could have originally been arranged, as to place the cards we must choose <math>n</math> of the original <math>13</math> spots to place the <math>n</math> cards from <math>1</math> to <math>n</math> (in that order). However, we must discard the case where the first <math>n</math> spaces are taken up by these cards, as then the cards would all be picked up in one pass. Hence, our desired number of ways to arrange the cards is
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| − | <cmath>\sum_{n=1}^{12} \left(\binom{13}{n} -1\right) = \left(\sum_{n=1}^{12} \binom{13}{n}\right) - 12 = \left(\sum_{n=0}^{13} \binom{13}{n}\right) - \binom{13}{0} - \binom{13}{13} - 12 = 2^{13} - 1 - 1 - 12 = \boxed{\textbf{(D)}\ 8178}.</cmath>
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| − | ==Solution 4 (Engineer's Induction, Risky)==
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| − | When we have <math>3</math> cards arranged in a row, after listing out all possible arrangements, we see that we have <math>4</math> ones: <math>(1, 3, 2), (2, 1, 3,) (2, 3, 1),</math> and <math>(3, 1, 2)</math>. When we have <math>4</math> cards, we find <math>11</math> possible arrangements: <math>(1, 2, 4, 3), (1, 3, 2, 4), (1, 3, 4, 2), (1, 4, 2, 3), (2, 1, 3, 4), (2, 3, 1, 4), (2, 3, 4, 1), (3, 1, 2, 4), (3, 1, 4, 2), (3, 4, 1, 2),</math> and <math>(4, 1, 2, 3).</math> Hence, we recognize the pattern that for <math>n</math> cards, we have <math>2^n - n - 1</math> valid arrangements, so our answer is <math>2^{13} - 13 - 1 = \boxed{\textbf{(D)}\ 8178}.</math>
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| − | == Video Solution By ThePuzzlr ==
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| − | https://youtu.be/p9xNduqTKLM
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| − | ~ MathIsChess
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| − | ==Solution by OmegaLearn Using Combinatorial Identities and Overcounting==
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| − | https://youtu.be/gW8gPEEHSfU
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| − | ~ pi_is_3.14
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| − | ==Solution==
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| − | https://youtu.be/ZGqrs5eg6-s
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| − | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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| − | == See Also ==
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| − | {{AMC12 box|year=2022|ab=A|num-b=18|num-a=20}}
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| − | {{AMC10 box|year=2022|ab=A|num-b=21|num-a=23}}
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| − | {{MAA Notice}}
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