56 lines
1.1 KiB
Ruby
56 lines
1.1 KiB
Ruby
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descr=%{
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There are exactly ten ways of selecting three from five, 12345:
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123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
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In combinatorics, we use the notation, 5C3 = 10.
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In general,
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nCr =
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n!
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r!(n−r)!
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,where r ≤ n, n! = n×(n−1)×...×3×2×1, and 0! = 1.
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It is not until n = 23, that a value exceeds one-million: 23C10 = 1144066.
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How many, not necessarily distinct, values of nCr, for 1 ≤ n ≤ 100, are greater than one-million?
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}
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remarks=%{
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C(n,r) == C(n,n-r)
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C(n,r) > X => forall m>n C(m,r)>X
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suppose r <= n-r
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5 x 4 x 3 x 2 x 1 5x4
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----------------------- = ------- = 10
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( 3 x 2 x 1 ) ( 2 x 1 ) 2x1
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C(n,r) = n x (n-1) x ... x (n-(n-r)+1) / r x (r-1) x ... x 1
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= n x (n-1) x ... x (r+1) / r x (r-1) x ... x 1
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}
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def C(n,r)
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if (r > n-r)
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r = n-r
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end
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top=1
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(n-r+1..n).each do |x|
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top *= x
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end
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down=1
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(2..r).each do |x|
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down *= x
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end
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return top/down
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end
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sum=0
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(1..100).each do |n|
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(1..n).each do |r|
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if C(n,r)>10**6
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sum+=1
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end
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end
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end
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puts sum
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