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Given

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ⁿC_r-1 = 36

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ⁿC_r+1= 126

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ⁿC _r  = 84

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ⁿC _r / ⁿC_r+1 = (r+1) / (n-r)

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84/126 = (r+1) / (n-r)

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2/3 = (r+1) / (n-r)

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2(n - r) = 3(r + 1)

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2n-2r = 3r+3

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2n - 5r - 3 = 0

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5r = 2n - 3

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ⁿC_r/ⁿC_r-1 = (n-r+1) / r

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84/36 = (n-r+1) / r

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7/3 = (n-r+1) / r

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3(n-r+1) = 7r

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3n-3r+3=7r

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3n-10r+3=0

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3n-2×5r+3=0

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Substitute 5r = 2n - 3

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3n-2(2n - 3)+3=0

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3n - 4n + 6 + 3 = 0

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n = 9

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5r = 2n - 3

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Substitute n = 9

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5r = 2*9 - 3

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5r = 15

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r = 3

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