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Question
Find n if 21C6n = `""^21"C"_(("n"^2 + 5))`
Solution
21C6n = `""^21"C"_(("n"^2 + 5))`
If nCx = nCy, then either x = y or x = n – y
∴ 6n = n2 + 5 or 6n = 21 – (n2 + 5)
∴ n2 – 6n + 5 = 0 or 6n = 21 – n2 – 5
∴ n2 – 6n + 5 = 0 or n2 + 6n – 16 = 0
If n2 – 6n + 5 = 0 then (n – 1)(n – 5) = 0
∴ n = 1 or n = 5
If n = 5 then
n2 + 5 = 30 > 21
∴ n ≠ 5
∴ n = 1
If n2 + 6n – 16 = 0 then (n + 8)(n – 2) = 0
n = – 8 or n = 2
n ≠ – 8
∴ n = 2
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