急! F5 排列與組合 4q1

22.
(a)
No. of combinations
= C(1,1) x C(1,1) x C(6,1)
= 1 x 1 x 6
= 6

(b)
No. of combinations
= C(8, 3) - 6
= 8!/3!5! - 6
= 56 - 6
= 50

(c)
No. of combinations
= C(6, 3) + C(2,1) x C(6,2)
= (6!/3!3!) + (2!/1!1!) x (6!/2!4!)
= 20 + 2 x 15
= 50


23.
(a)
Number of triangles
= C(n,3)
= n!/3!(n - 3)!
= n(n - 1)(n - 2)/6

Number of quadrilaterals
= C(n,4)
= n!/4!(n - 4)!
= n(n - 1)(n - 2)(n - 3)/24

Total number of triangles and quadrilaterals
= n(n - 1)(n - 2)/6 + n(n - 1)(n - 2)(n - 3)/24
= (1/24)n(n - 1)(n - 2)[4 + (n - 3)]
= n(n - 1)(n - 2)(n + 1)/24

(b)
n(n - 1)(n - 2)(n + 1)/24 = 35
n(n - 1)(n - 2)(n + 1) = 840
n(n - 1)(n - 2)(n + 1) = 6 x 5 x 4 x 7
n(n - 1)(n - 2)(n + 1) = 6(6 - 1)(6 - 2)(6 + 1)
Hence, n = 6