N choose 4 = N choose
我計到 n(n-1)(n-2)(n-3) n(n-1)
---------------------- = ------------
24 2
就不懂了
PLZ給詳細過程
MATH M2
2015-08-01 6:50 am
回答 (3)
2015-08-06 6:53 am
無最佳解答
2015-08-06 1:04 am
2015-08-01 7:24 am
C(n,4) = C(n,2)
n! / [(n-4)! 4!] = n! / [(n-2)! 2!]
n (n-1) (n-2) (n-3) (n-4)! / [(n-4)! 4!] = n (n-1) (n-2)! / [(n-2)! 2!]
n (n-1) (n-2) (n-3) / 24 = n (n-1) / 2
n (n-1) (n-2) (n-3) = 12 n (n-1)
n (n-1) (n-2) (n-3) - 12 n (n-1) = 0
n (n-1) [ (n-2) (n-3) - 12 ] = 0
n (n-1) ( n² - 5n + 6 - 12 ) = 0
n (n-1) ( n² - 5n - 6 ) = 0
n (n-1) (n-6) (n+1) = 0
n = 0 or 1 or 6 or -1
Since C(n,4) implies n ≥ 4
∴ n = 6
2015-07-31 23:31:21 補充：
Property：C(n,r) = C(n,n-r)
If C(n,4) = C(n,2),
Let C(n,r) = C(n,4) ⇒ r = 4
∴ C(n,n-r) = C(n,2) ⇒ n - 4 = 2 ⇒ n = 4 + 2 = 6
or
Let C(n,r) = C(n,2) ⇒ r = 2
∴ C(n,n-r) = C(n,4) ⇒ n - 2 = 4 ⇒ n = 2 + 4 = 6
∴ n = 6
n! / [(n-4)! 4!] = n! / [(n-2)! 2!]
n (n-1) (n-2) (n-3) (n-4)! / [(n-4)! 4!] = n (n-1) (n-2)! / [(n-2)! 2!]
n (n-1) (n-2) (n-3) / 24 = n (n-1) / 2
n (n-1) (n-2) (n-3) = 12 n (n-1)
n (n-1) (n-2) (n-3) - 12 n (n-1) = 0
n (n-1) [ (n-2) (n-3) - 12 ] = 0
n (n-1) ( n² - 5n + 6 - 12 ) = 0
n (n-1) ( n² - 5n - 6 ) = 0
n (n-1) (n-6) (n+1) = 0
n = 0 or 1 or 6 or -1
Since C(n,4) implies n ≥ 4
∴ n = 6
2015-07-31 23:31:21 補充：
Property：C(n,r) = C(n,n-r)
If C(n,4) = C(n,2),
Let C(n,r) = C(n,4) ⇒ r = 4
∴ C(n,n-r) = C(n,2) ⇒ n - 4 = 2 ⇒ n = 4 + 2 = 6
or
Let C(n,r) = C(n,2) ⇒ r = 2
∴ C(n,n-r) = C(n,4) ⇒ n - 2 = 4 ⇒ n = 2 + 4 = 6
∴ n = 6
收錄日期: 2021-04-16 17:07:37
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20150731000051KK00081