Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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2 ·10
15
n
1
n
2
n
1
n
2
k
n
1
n
2
. . . k n
k
k n
1
n
2
···n
k
. . .
5 ·6 ·6 = 180
k k > 1 n
k k
n
k n
k n
n k
n n
n
k n
k n
n k
k n
k n
n k
k n
k
n k
n k
n k A
k
n
k
n
A
k
n
= n(n −1) ···(n − k + 1).
n P
n
n
P
n
= A
n
n
= 1 ·2 · 3 ···n.
n
n! n
P
n
= n!, A
k
n
=
n!
(n − k)!
n k C
k
n
n k
k
C
k
n
=
A
k
n
P
k
=
n!
k!(n − k)!
.
n k
A
k
(n)
n
k
n k
C
k
(n)
n + k − 1
k
C
k
(n)
= C
k
n+k−1
.
k n m
A
P (A) =
m
n
k
.
k n m
A
P (A) =
m
A
k
n
.
n k m
A
P (A) =
m
C
k
n
.
n = A
4
(6)
= 6
4
A
m = A
4
6
P (A) =
A
4
6
6
4
=
6 · 5 · 4 · 3
6
4
=
5
18
≈ 0, 28. ¤
n = 10!
A
m = 9! · 2
P (A) =
9! · 2
10!
= 0, 2. ¤
n = C
3
10
A
i
i = 0, 1, 2, 3
i 3 −i i
C
i
6
C
0
n
:= 1 3 − i
C
3−i
4
A
i
m = C
i
6
C
3−i
4
P (A
0
) =
C
0
6
· C
3
4
C
3
10
=
1
30
,
P (A
1
) =
C
1
6
· C
2
4
C
3
10
=
3
10
,
P (A
2
) =
C
2
6
· C
1
4
C
3
10
=
1
2
,
P (A
3
) =
C
3
6
· C
0
4
C
3
10
=
1
6
.
¤
A
P (A) = .
x
y
(x, y)
0 6 x 6 60 ( ), 0 6 y 6 60 ( ),
A
|x − y| 6 20.
P (A) =
S
S
=
60
2
− 40
2
60
2
=
5
9
.
5
9
¤
2a
2l l < a
x
α
(α, x)
0 6 x 6 a
0 6 α 6 π
A
x 6 l sin α.
S = aπ,
S =
π
Z
0
l sin α dα = 2l,
P (A) =
2l
aπ
. ¤
P
∗
= ,
P
∗
≈ P (A),
P
∗
≈
2l
aπ
π ≈
2l
aP
∗
.
π
π = 3, 14159265.
π
π