# Variations 3rd class

From how many elements we can create 13,800 variations 3rd class without repeating?

Result

n =  25

#### Solution:

$V_3(n) = \dfrac{ n! }{ (n-3)! } = \dfrac{ n(n-1)(n-2)(n-3)! }{ (n-3)! } = n(n-1)(n-2) = 13800 \ \\ n(n-1)(n-2) = 13800 \ \\ n \approx \sqrt[3]{13800} = 23.986 \ \\ 22 \le n \le 26 \ \\ V(3,22) = 22 . 21 . 20 = 9240 \ \\ V(3,23) = 23 . 22 . 21 = 10626 \ \\ V(3,24) = 24 . 23 . 22 = 12144 \ \\ V(3,25) = 25 . 24 . 23 = 13800 \ \\ V(3,26) = 26 . 25 . 24 = 15600 \ \\ \ \\ n = 25$

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