# Find n, if 21C6n=21C(n2+5) - Mathematics and Statistics

Sum

Find n, if ""^21"C"_(6"n") = ""^21"C"_(("n"^2 + 5)

#### Solution

We have ""^21"C"_(6"n") = ""^21"C"_(("n"^2 + 5)
∴ 6n = n2 + 5
∴ n2 – 6n + 5 = 0
∴ (n – 5)(n – 1) = 0
If n = 5,  ""^21"C"_(6"n") = ""^21"C"_30
Which is possible,
∴ n = 5 is discarded.
Also ""^21"C"_(6"n") = ""^21"C"_(("n"^2 + 5)
∴ ""^21"C"_(21 - 6"n") = ""^21"C"_(("n"^2 + 5)
∴ n2 + 6n – 16 = 0
∴ n + 8 = 0 or n – 2 = 0
∴ n = – 8 or n = 2
But n ∈ N
∴ n = 2

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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board
Chapter 6 Permutations and Combinations
Exercise 6.7 | Q 3 | Page 90