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| − | ==Problem==
| + | #redirect [[2022 AMC 10A Problems/Problem 22]] |
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| − | Suppose that 13 cards numbered 1, 2, 3, . . . , 13 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.
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| − | For how many of the 13! possible orderings of the cards will the 13 cards be picked up in exactly
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| − | two passes?
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| − | ==Solution==
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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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| − | \textbf{Case 1:}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). Finally, card <math>m</math> goes into whatever slot is remaining. Therefore in this case there are <math>\binom{n}{n-1}</math> possibilities.
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| − | Solution in Progress
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| − | ~KingRavi
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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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