數學:P和C

Question

1.Pn+2取4:P2n取3=3:2,求Cn取6=?(請寫出計算過程)
2.C2n取n-1:C2n-2取n=132:35,n屬於N,n=?(請寫出計算過程)
3.若C1取0+C2取1+C3取2+...+Ck取k-1=190,則k=?(請寫出計算過程)
4.Sigam頂數是48,底數是K=0*C50-k取48-k=?(請寫出計算過程)

Answer 1

1. P(n+2, 4) : P(2n, 3) = 3:2
=> [(n+2)(n+1)(n)(n-1)] : [(2n)(2n-1)(2n-2)] = 3:2
=> [(n+2)(n+1)(n)(n-1)] : [4(n)(2n-1)(n-1)] = 3:2
=> [(n+2)(n+1)] : [4(2n-1)] = 3:2
=> 2(n+2)(n+1) = 12(2n-1)
=> n2 - 9n+ 8 = 0
=> n = 8 or 1 (不合)
C(n, 6) = C(8, 6) = C(8, 2) = 28
2. C(2n, n-1) : C(2n-2, n) = 132 : 35
=> (2n)! / {[2n-(n-1)]!*(n-1)!} : (2n-2)! / {[(2n-2) - n]! * n!} = 132:35
=> (2n)! / [(n+1)!*(n-1)!] : (2n-2)! / [(n-2)! * n!] = 132:35
=> [(2n)(2n-1)] / [(n+1)(n-1)] = 132 / 35
=> 70n(2n-1) = 132(n2 - 1)
=> 4n2 - 35n + 66 = 0
=> n = 6 or 11/4 (不合)
=> n = 6

3. C(m, m-1) = C(m, 1) = m
=> 原題 = 1 + 2 + ... + k = 190
=> k(k+1)/2 = 190
=> k2 + k - 380 = 0
=> k = 20 or -19 (不合)
=> k = 20

4. 題中有個 * 是什麼意思?
假設題目是 Σ48k=0 C(50-k, 48-k) = ?
C(50-k, 48-k) = C(50-k, 2) = (50-k)(49-k)/2
Σ48k=0 C(50-k, 48-k)
= Σ48k=0 (50-k)(49-k)/2
= (1/2)Σ48k=0 (k2 - 99k + 2450)
= (1/2){[48*(48+1)(2*48+1)/6] - 99[48*(48+1)/2] + 49*2450}
= (1/2){38024 - 116424 + 120050}
= 20825
如果有問題, 請來函討論. 不然, 我可能會錯失你再補充的疑點.