Page 104 - MATINF Nr. 13-14

104 M.N. Popescu

deci
2
1
0
card F n + card D n = C · D n + C · D n−1 + C · D n−2 + . . . + C n−2 · D 2 + 1.
n n n n
Cum S n = D n ∪ F n , D n ∩ F n = Ø s , i card S n = n!, obt , inem c˘a
1
0
2
n−2
n! = C · D n + C · D n−1 + C · D n−2 + . . . + C n · D 2 + 1.
n
n
n
Cum D 1 = 0, notˆand D 0 = 1, relat , ia anterioar˘ devine
a
2
0
1
n
n! = C · D n + C · D n−1 + C · D n−2 + . . . + C n n−2 · D 2 + C n n−1 · D 1 + C · D 0 .
n
n
n
n
k
Folosind C = C n−k , rescriem relat , ia anterioar˘ sub forma
a
n
n
n
X
k
n ! = C D k .
n
k=0
Folosim punctul 5.d pentru u k = k!, v k = D k s , i obt , inem
n n n n−k n k
X n−k k X n−k n! X (−1) X (−1)
D n = (−1) C · k! = (−1) · k! = n! = n! .
n
k! · (n − k)! (n − k)! k!
k=0 k=0 k=0 k=0
7. a. Probabilitatea uniform˘a pe D n este
card (A)
P (A) = , ∀A ⊂ D n .
card (D n )
Din punctele 4 s , i 6 avem
n−1
card {σ ∈ D n : ε (σ) = 1} − card {σ ∈ D n : ε (σ) = −1} = (−1) (n − 1) ,
n k
X (−1)
card {σ ∈ D n : ε (σ) = 1} + card {σ ∈ D n : ε (σ) = −1} = n! ,
k!
k=0
din care obt , inem
" n k #
1 n−1 X (−1)
card {σ ∈ D n : ε (σ) = 1} = (−1) (n − 1) + n! ,
2 k!
k=0
" n #
1 n−1 X (−1) k
card {σ ∈ D n : ε (σ) = −1} = −(−1) (n − 1) + n! .
2 k!
k=0
Prin urmare

n
P (−1) k n−1
n! − (−1) (n − 1)
card {σ ∈ D n : ε (σ) = −1} 1 k=0 k!
P (Y n = −1) = = · ,
n
card (D n ) 2 P (−1) k
n!
k!
k=0
n
P (−1) k n−1
n! + (−1) (n − 1)
card {σ ∈ D n : ε (σ) = 1} 1 k!
P (Y n = 1) = = · k=0 .
n
card (D n ) 2 P (−1) k
n!
k!
k=0