Combination

13Cn+2 = 13C3n+1
13! / [13 - (n + 2)]!(n + 2)! = 13! / [13 - (3n + 1)]!(3n + 1)!

[13 - (n + 2)]! = (3n + 1)! or (n + 2)! = (3n + 1)!
11 - n = 3n + 1 or n + 2 = 3n + 1
4n = 10 or 2n = 1
n = 2.5 (rejected) or n = 0.5(rejected)

Hence, there is no solution.

2011-10-14 15:35:05 補充：
其實可以看成：
13!/(13-a)!a! = 13!/(13-b)!b!

由於兩分數及其分子均相等，故其分母必相等：
(13-a)!a! = (13-b)!b!

要上式兩邊相等，則 (13 - a)! = (13-b)! 或 (13 - a)! = b!

而 a! = b! 與 (13 - a)! = (13-b)! 得相同結果。
a! = (13 - b)! 與 (13 - a)! = b! 得相同結果。

C is combination.......nCr

13C(n+2) = 13C(3n+1)
first, both sides are divided by 13C

n+2 = 3n+1
1= 2n
n= 1/2