document.write( "Question 968110: how many ways there are to remove three elements from the sequence 1 4 2 5 7 3 8 9, so that the remaining from an increasing sequence\r
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Algebra.Com's Answer #591672 by KMST(5328)\"\" \"About 
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1 4 2 5 7 3 8 9 is a sequence of 8 digits (I assume there was no 6).
\n" ); document.write( "The digits 1, 2, and 3 are followed by digits that are greater.
\n" ); document.write( "The digit 8 and 9 are not followed by any digit with lesser value.
\n" ); document.write( "The inversions (deviations from increasing order) happen when
\n" ); document.write( "4 is followed by 2,
\n" ); document.write( "4 is followed by 3,
\n" ); document.write( "5 is followed by 3, and
\n" ); document.write( "7 is followed by 3.
\n" ); document.write( "So, there are four ordered pairs that make the sequence deviate from the increasing order:
\n" ); document.write( "(4,2), (4,3), (5,3), and (7,3).
\n" ); document.write( "Removing the set of elements {4,5 7} is \"red%281%29\" way to eliminate all four inversions.
\n" ); document.write( "Can we keep the number 4?
\n" ); document.write( "It is involved in 2 inversions: (4,2), and (4,3).
\n" ); document.write( "If we want to keep 4, we need to remove at least two numbers: 2 and 3.
\n" ); document.write( "If we are going to remove 3 numbers, along with 2 and 3, but we want to keep 4,
\n" ); document.write( "we can remove any of the other \"8-3=5\" numbers,
\n" ); document.write( "along with 2 and 3.
\n" ); document.write( "That makes \"red%285%29\" different sets of 3 numbers.
\n" ); document.write( "Of course, if we remove number 4, we eliminate the inversions (4,2), and (4,3),
\n" ); document.write( "and we do not need to remove the entire set {4,5 7}.
\n" ); document.write( "We can keep 5 and/or 7, as long as we remove number 3,
\n" ); document.write( "which would eliminate the other two inversions: (5,3), and (7,3).
\n" ); document.write( "How many ways can we chose a set of 3 numbers to remove, including 4 and 3?
\n" ); document.write( "Along with 4 and 3, we can remove any of the other \"8-2=6\" number,
\n" ); document.write( "so we can do that \"red%286%29\" different ways.
\n" ); document.write( "We found \"red%281%29%2Bred%285%29%2Bred%286%29=highlight%2812%29\" ways
\n" ); document.write( "\"to remove three elements from the sequence 1 4 2 5 7 3 8 9, so that the remaining from an increasing sequence\".
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