Удод-Андриенко методичка
.pdf"! " ,, ": ) ) "' " % " &' % " ) % " ) 3; b) ) "' " % " &' % " ) & ') 3 " ) 3; ) ) & ') 3 " ) 3 &'- % " " ! % " % ".
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"! " ,, ": ) " % "+ $% " $% "! ) % " ); b) " % "+ $% " $% "! ) " ) "' "! " ); ) " % "+ $% " $% "! ) ' 1 +( " )+(; d) ) )" ' " ) $% "$% "! % " % ".
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"! " ,, " ' % )+ "+( 5 ' ): ) "! $ + 5 '; b) " '- , 5 '.
1.4.2. 5&$ 2$# " 2
" 1 " ") $*, & % +( &' % + ) 0, " '+ 2 " ) "! 1 - " )
1 % , , 2 "). ' & ' % ") )+ -
) ' 0" ", & !1 ) , &' % 0
(4), " &' 03" ' + $" '. ' ) % $
' &' "' + 1 (.
" 2"$ 1+) 3" 0 $*, " ) + 1 %- ( " ( 2 n ' 1 +( - " ), " '+ " 3" 0 2%$ " ! & '0% ' & 2 0 - " ). $4 &- ' " ) ' )
Pn = n! , |
(5) |
,% n! = 1 B 2 B 3 B … B n .
)"$*". ) 0 1 & ! n! " " 0 «- " '-
». &' % 3 & , 3" 0! = 1.
$ 24. ! ' 1 +( "' (1 +( 2 - " ) "! 1 * ' 1, 2, 3?
. ' (1 + , " ) + 1 * ' 1, 2, 3,
&' % " ) 03" $ $* " ) + 1 % ( " ( 2 "' ( - " ), " '+ " 3" 0 2% $ " ! & '0%
' & 2 0 - " ), ". . 0) 03" 0 & ' " ) 1 "' (
- " ). ' (5) ") " ( $* ' )
P3 = 3! = 1C 2 C 3 = 6 . ' -" : 123, 132, 213, 231, 312, 321.
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$ 25. '0% )+ " & 0 7 " ) ' &'- % 0 " 0 2'$ . ! ' 1 +( ) ' " ) 2 '$! ) &' -" ) 1 2?
. : 2%+ ) ' " 2 '$! ) " " 0 " %' , ( )- ' " ) " ! & '0% " ) ', ". . 0) 0 " 0 & '- " ) 1 7 - " ). ' (5) ") " ( $ -
* ' )
P7 = 7! = 1C 2 C 3 C 4 C 5 C 6 C 7 = 5040.
* "$ 1+) 3" 0 $*, " ) + 1 n ' 1-
+( - " ) & m - " ), " '+ " 3" 0 2% - $ " ! " ) - " ), ". . ( "0 $+ % - ". $- 4 " ' )
Cnm = |
n! |
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)"$*". " 0( & '0% ' & 2 0 - " )
) 2.
$ 26. ! &$ 2 )+$' "! & ' * ' 1 "' ( * ': 1, 2, 3, & '0% ' & 2 0 * ' ) "' & '+ ) 2?
. '+, " ) + 1 * ' 1, 2, 3, &' % " ) 03" - $ $* 1 "' ( - " ) & %) - ", " '+ " - 3" 0 2% $ " ! " ) - " ), ". . 0) 03" 0 - " 0 1 "' ( - " ) & %). ' (6) ") " (
$* ' )
C32 = |
3! |
= |
1× 2 × 3 |
= 3 . |
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2!×1! |
1× 2 ×1 |
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' -" & '+: (1, 2), (1, 3), (2, 3)
$ 27. 5 ( " " ' ' ") 3" 16 ). !-
& '" % 2 $+"! +,' ) " ' ', 2% 3$+ %) 0 " % 2 $+"! +,' % & '" 0?
. : 2% 0 & '" 0 ,' " 0 %) 0 " 1 16 "-
" 0 " %' , ( " ! " ) & ' " ), ". . &' % " )-
0 " $ " 1 16 - " ) & 2. ( ( % & ' (6):
C 2 |
= |
16! |
= 14! ×15 ×16 = 15 ×8 = 120 . |
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16 |
14!× 2! |
14!×1× 2 |
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",$.$ 1+) 3" 0 $*, " ) + 1 n ' 1-
+( - " ) & m - " ), " '+ " 3" 0 2% -
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$ $ " ) - " ), $ ( & '0%. $4 ' 1- 4 ' )
Am = |
n! |
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(7) |
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n |
(n - m)! |
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$ 28. ! ' 1 +( %) 1 +( 2 "- ) "! 1 * ' 1, 2, 3?
. ) 1 + , " ) + 1 * ' 1, 2, 3, &' %- " ) 03" $ $* 1 "' ( - " ) & %), " '+ "-
3" 0 2% $ $ " ) - " ), $ ( & '0%, ". . 0) 03" 0 ' 1 4 0 1 "' ( - " ) & %). '
(7) ") " ( $* ' )
A2 |
= |
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3! |
= 3! = 2 × 3 = 6 . |
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3 |
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- 2)! |
1! |
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' -" : 12, 13, 21, 23, 31, 32.
$ 29. & % , % 0 " " 1 5 ' ). "
) ' " ) ' & 0 &' )+$' 1 11 % * & .
. : 2%+ ) ' " ' & 0 &' % " ) 0 " $ - $* 3 1 11 % * & & 5, " 34 3 0 " %' , ( ) ' " ) ( $*) $ " ) % * & , $ & '0% ( %- ) 0 ( ", %' ,), ". . 0) 0 " 0 ' 1 4 1 11 - -
" ) & 5. 8 ) ' " ) ' &, ". . ' 1 4 1 11 & 5, ( % & ' (7)
A5 |
= |
11! |
= 11! = 6!× 7 ×8 × 9 ×10 ×11 = 7 ×8 × 9 ×10 ×11 = 55440 . |
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11 |
(11- 5)! |
6! |
6! |
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' ' 5 1 % $" ' " & !1 3" 2 -
% 34 &' ) .
" & +$$ . " '+ $< " % 2 " $+"! )+$' 1 ) & " $< " ) m &$ , %' , $< " & 2 " $+"! )+$' n &$ , " )+$' "! $ %, $ & 2 m + n
&$ .
" & , (. $< " % 2 )+$' "! 1 )- & " $< " ) m &$ & 2% , " , )+$'$< " & 2 )+$' "! n &$ , " & ' $< " ) (%, &) ) - 1 & '0% 2 " $+"! )+$' m-n &$ .
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$ $ 6 $+& 2$# " 2 * &
!
$ 30. ) " ) 1 '", 2% 1 "- '+( & % $). " '" 5 ) 3" )+ 3"
, % $1 ) 1)' " & %. " ) ' 0" "! " ,, " $)+ )+ 3" 0 ) & '0% 1 % , ): ) :, b) .
. a) = , 1. &+" – )+ &0" '" $ - ) ) & '0% & % $1 ) 1)' ".
= , 2. ' "' ") - " ' +( $+" 6 " " 1 )- ) 1 2 +( $*, " ) +( 1 % ( " ( 2 $) «:», « », « », « », « ». :$* " 3" 0 2% $ " ! & '0% ' & 2 0 $). % ) " !, $* (- -
" ' + $+" 0) 0) 03" 0 & ' " ) 1 5 $), 1 ", & ' (5) | 6 | = P5 = 5! = 120.
= , 3. " ' 34 $+" % = {$)+ )+ "+ ) & '0%) :} = {:} " " 1 % ") , - " '- , $+" 0 – & ' " ) «:». -" | A | = 1.
= , 4. ) - " ' + $+" 0 ' ) ) ' 0" +, "
2 ) & !1 ) "! 0 ' ) ' 0" " (4).
$' 1, (%) = | A | / | 6 | = 1 / 120.
b)= , 1. &+" – )+ 5 " '" $) )
& '0% & % $1 ) 1)' ".
= , 2. ' "' ") - " ' +( $+" 6 " " 1 )-
) 1 2 +( $*, " ) +( 1 % ( " ( 2 $) « », « », « », « », « », « ». :$* " 3" 0 2% $
" ! & '0% ' & 2 0 $). % ) " !, $*
(- " ' + $+" 0) 0) 03" 0 & ' " ) 1 6 $), 1- ", & ' (5) | 6 | = 6! = 720.
= , 3. " ' 34 $+" & = {$)+ )+ "+ ) & '0%) } = { }. 0 " ,, "$+ & % " "! $4
") & ' " ), &' " '+( & " 0 ) « », $( % "!, " & )" '034 0 $)+ « » « » 2 &' 1) ! + $' 1 & ' " ) 0"! 2% $ , &' -" ) 1 " 0. ' $)+ « » 2 ' 1 " "! & '- ), "' "! &0" " ( 3! = 6 &$ . ) $)+ « » 2-
' 1 " "! )" ' ") '" " ( 2! = 2 &$ . : 2- % 3 & ' " ) 1 $) « » 2 $' ) "! 2% &- ' " ) 1 $) « ». % ) " !, $4 ") & '- " ), &' " '+( & " 0 ) « », , &'- ) &' 1) % 0 ' ) 6 - 2 = 12, " "! | & | = 12.
24
= , 4. ) - " ' + $+" 0 ' ) ) ' 0" +, "
2 ) & !1 ) "! 0 ' ) ' 0" " (4).
$' 1, (&) = | & | / | 6 | = 12 / 720 = 1 / 60.
$ 31. ' 4 ' +( 5 $ +( 5 ' ). 1 ' + - + $' 1 )+ 3" 3 5 '. " ) ' 0" "! " ,, " ' % )+ "+( 5 ' ):
a)) 5 '+ $ +;
b)) 5 '+ % , *) ";
c)2 $ +( 5 ';
d)!5, 2 $ +( 5 ';
e)( "0 $+ % $ + 5 '.
. = , 1. &+" – )+ + $' 1 3 5 ' ) 1 ' +.
= , 2. ' "' ") - " ' +( $+" 6 " " 1 )- ) 1 2 +( $*, " ) +( 1 % )0" 5 ' ) & "', - " '+ " 3" 0 2% $ ( "0 $+ % - ". '0% )+ 0 5 ' ) ) 2. % ) " !, $* (- -
" ' + $+" 0) 0) 03" 0 ) ) 1 2 + " 0 & 3 1 9 5 ' ). ( ( % & ' (6):
| W |= C93 = |
9! |
= 6!× 7 ×8 × 9 = 7 ×8 × 9 |
= 7 × 4 × 3 = 84 . |
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3! 6! 1× 2 × 3× 6! 1× 2 × 3 |
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= , 3, 5 , 4. a) " ' 34 $+" % = {) )+ "+ 5- '+ – $ +}. ' $ +( 5 ' 2 )+ "! 1 &0" $ +(
C53 = |
5! |
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= 10 &$ . % ) " !, | A | = 10, (%) = |
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3!×2! |
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=| A | / | 6 | = 10 / 84 = 5 / 42.
b)" ' 34 $+" & = {) )+ "+ 5 '+ % , *)-
"}. ' 5 ' % , *) " , " $+"! $ ' +, $ $ +.
' $ +( 5 ' 2 )+ "! 1 &0" $ +( C53 = 3!5×!2! = 10 &- $ . ' ' +( 5 ' 2 )+ "! 1 "+' ( ' +(
C43 = 3!4×!1! = 4 &$ . 1 ' + )+ 3" 3 5 ', " &-
0) () ' 1 !" " )+ 0) 3$ $* 1 3 $ +( 5 ' ) 3 " & 0) 3$ $* 1 3 ' +( 5 ' ).-" , &' ) + $4 $* $ ,-
&' 0" +( $+" 3 & $% " ' ) 10 + 4 = 14. % ) " !, | & | = 14, (&) = | & | / | 6 | = 14 / 84 D 0.167.
25
c) " ' 34 $+" ! = {' % )+ "+( 5 ' ) " 0
' ) 2 $ +( 5 '} = {' % )+ "+( 5 ' ) " 0 ' ) 2 $ - +( 1 ' + 5 '}. ) $ +( 5 ' 2 )+ "! 1 &0" $ +(
C52 = 25!!3! = 10 &$ . % ' + 5 ' 2 )+ "! 1 -
"+' ( ' +( C41 = 1!43!! = 4 &$ . : 2% " 1 2 $ -
+( 5 ' ) 2 $' ) "! 2%+ " 1 1 '- , 5 '. % ) " !, , &' ) &' 1) % 0 $4 -
") |
$* |
$ , &' 0" +( $+" 3 |
! $% " |
' ) |
10 - 4 = 40, |
" "! |
| ! | = 40. -" |
(!) = | ! | |
/ | 6 | = |
=40 / 84 = 10 / 21.
d)" ' 34 $+" D = {' % )+ "+( 5 ' ) " 0
!5, 2 $ +( 5 '}. $+" D " & ", $% " )+ " 1 $ + 5 ' 2 ' +(, $% " )+ " 0 $ +( 5 ' ) 3 ' +(.
% $ + 5 ' 2 )+ "! 1 &0" $ +( C51 = 1!54!! = 5 &-
$ . ) ' +( 5 ' 2 )+ "! 1 "+' ( ' +(
C42 = 2!4!2! = 6 &$ . : 2% " 1 1 $ , 5 ' 2-
$' ) "! 2%+ " 1 2 ' +( 5 ' ). %- ) " !, , &' ) &' 1) % 0 $4 ") &-
$) )+$' |
% , $ , |
%) ( ' +( |
5 ' ) $% " ' )- |
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5 - 6 = 30. |
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' |
' +( 5 ' 2 |
)+ "! 1 |
"+' ( ' +( |
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C43 = |
4! |
= 4 |
&$ , " ") " ") " 3, ,% )+ " 0 |
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3!×1! |
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$ +( 5 ' ) 3 ' +(.
$4 $*, $ , &' 0" +( $+" 3 D ,
&' ) + $% " ' ) $*, ") " ") 34 (
-" %) )1 3 34 0 (& ')+ – )+ "
' ) 1 $ + 2 ' +( 5 ', )" ' – )+ " 0 $ +( 3 '- +( 5 '), " "! | D | = 30 + 4 = 34. % ) " !, (D) =
=| D | / | 6 | = 34 / 84 = 17 / 42.
e)" ' 34 $+" ) = {' % )+ "+( 5 ' ) " 0
( "0 $+ % $ + 5 '}. ,% , ) ' " 0 &' «( "0 $+ %» &' 0" & ' ( % "! &' " ) & 2 $+" 3. 0 -
26
$+" 0 ) &' " ) & 2 + $% " $+" E = {' % )+ "+( 5 ' ) " $ +(} = {) "' )+ "+( 5 ' – ' +}. ' ' +(
5 ' 2 )+ "! 1 "+' ( ' +( |
C43 = |
4! |
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= 4 &$ . |
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3!×1! |
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% ) " !, | |
E |
| = 4. -" ( |
E |
) = | |
E |
| / | 6 | = 4 / 84 = 1 / 21. |
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" 3%, & !1 0 ) ") 3 ) ' 0" ", & P()) = 1 –
( E ) = 1 – 1 / 21 = 20 / 21.
$ 32. ' $' " , ' $" 1$+ %) & % * '+ $' ( %, & 0 " !, " -" *- '+ " + ' 1 +. " ) ' 0" "! " ,, " ' $' &' ) !.
. = , 1. &+" – $' %) ( & % ( * ' " -
, '.
= , 2. ' "' ") - " ' +( $+" 6 " " 1 )- ) 1 2 +( $*, " ) +( & %) - " (& %) * '+) 1 &0" - " ) (&0" " +( * '), " '+ " - 3" 0 2% $ $ " ) - " ), $ ( & '0%.% ) " !, $* (- " ' + $+" 0) 0) 03" 0 ' 1-
4 0 & 2 |
1 5 * '. ( ( % & ' (7): |
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| W | = A2 |
= |
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5! |
= 5! = 3!× 4 ×5 = 4 ×5 = 20 . |
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5 |
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(5 |
- 2)! |
3! |
3! |
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" ", " $+ * '+ $+ ' 1 +, " $*
!10 $+ $+ 1) "! ' 1 4 0, " 1 " +( * ' (1, 3, 5, 7, 9) ) 1 2 )+$' "! %) % )+. %$+ - $* 1+) 3" 0 ' 1 4 0 & )" ' 0, % 0 ( ' (7) ) '.
= , 3. " ' 34 $+" % = {%) & % * '+ - $' + ) '} " " 1 % , - " ' , $+" 0 (". . -
' " + |
" + ' 0) 0 " 0 |
% ") +). |
-" |
| A | = 1. |
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= , 4. |
' |
) ' 0" " (4) |
& |
(%) = | A | / | 6 | = 1 / 20.
)"("*
35. 0" " $' ' & 2 & ) -
& '0%. : ) ) ' 0" "! " ,, " , " 0" ) &' ) ) & '0% ' * " ) (" 1 % 5)?
27
36.) " ) 1 '", 2% 1 " '+( & % $). " '" 5 ) 3" )+ 3" , % $1 ) 1- )' " & %. " ) ' 0" "! " ,, " $)+ )+ 3" 0 ) & '0% 1 % , ):
a)9 9 :;
b):;
c)99;
d)/;
e)9 E9.
37." ) ' 0" "! " ,, " 1 4 ,, ' & 2 +( ) -
& '0%, 2 &' % + , 2 " 0 '0%.
38.6 ) + $' 1 ' 2 ) 3" 0 1 ' , + " .
" ) ' 0" "! " ,, " %) &' % +( ) 2 " 0
'0%.
39.6 ) + $' 1 ' 2 ) 3" 0 . - " ) ' 0" "! " ,, " %) &' % +( ) 2 " 0 '0- %.
40.' %! ) $ 3 + $' 1 ) " 6 2 4 2 2. " ) ' 0" "! " ,, " 2% 2 $% " "- 0"! %) 2 4 +.
41*. n ) + $' 1 ' 2 ) 3" 0 1 ' , + " .
" ) ' 0" "! " ,, " %) &' % +( ) 2 " 0
'0%.
42*. n ) + $' 1 ' 2 ) 3" 0 .
" ) ' 0" "! " ,, " %) &' % +( ) 2 " 0
'0%.
43.' 3 $ +( 7 ' +( 5 ' ). " ) ' 0" "! " ,, " )+ "+ % 1 ' + %) 5 ' 2 " 0 ' +.
44.04 100 % " , 1 ( 10 $' ) +(. % 1) - + "+' % " . " ) ' 0" "! " ,, " ' % 1) +(
% " : ) " $' ) +(; $) " " % '" +(.
45." % " 1 " 20 1 25 ) &' ) &' ,' +. " ) ' 0" "! " ,, " " % " 1 " &' % 2 + -1 " ' "' )- &'.
46.' 5 $ +( 7 ' +( 5 ' ). 1 ' + % )+ 3" 0 %) 5 '. " ) ' 0" "! " ,, " )+ "+ 5 '+ $% " ' 1 +( *) " ).
47.$' , % 4 2 4 + 3 2 +. ' % ) $' , %+ '- 1+,'+) 3" 0 4 $ " ) " "'. : ) ) ' 0" "! " ,, " ' %$ % " $ " ) 2 4 2 2 " 0 & ' )?
28
48.1 % 0" $ " ) )+ ,'+5 + 0) 03" 0 %). 8 ' ) )- ' 0" "! " ,, " ' % )10"+( % &0" $ " ) " ! % )+ ,'+5 +?
49.,' && 20 " % " ), ' % " '+( 8 " ). &
% "$' + 9 " % " ). " ) ' 0" "! " ,, " ' % "$' +( " % " ) &0"! " ).
50.& '" 1 10 % " 7 " % '" +(. " ) ' 0" "! " ,, " ' % 6 )10"+( % % " 4 " % '" +(.
51.# $'4 3" 0 10 % " , " 34 ( 0 %' , " %' ,, 1 ( "+' – & ') ,, & %) – )" ' ,, "' "! , ") '- " , ) % ). " ) ' 0" "! " ,, " ' % 5 " )10"+( %- )' % " "' 2 " 0 & ') , ) %, %) – )" ' , % – "' "! ,.
52.& " 4 ' +( 6 1 +( % * ). " ) ' 0" "! " ,, " )+ "+ % 1 & " 3 % * 2 " 0 % , *) ".
53.' 3 $ +( 4 ' +( 5 '. 1 ' + % )+ 3" %) 5 '. : $+" $ ) ' 0" % = {)+ "+ 5 '+ % , *) "} & = {)+ "+ 5 '+ ' 1 +( *) " )}?
54.& '" 1 20 1% 4 $' ) +(. " ) ' 0" "! " ,, " ' % 5 )+$' +( 1% $ % , $'- ) ,.
55.' 10 $ +( 9 ' +( 5 ' ). 1 ' + )+ 3" %
55 ' ). " ) ' 0" "! " ,, " ' % )+ "+( 5 ' ) "!
$ + ' +.
56." ' 100 $ " ), 1 " '+( 20 )+ ,'+5 +(. # " " ' & & " 3 $ ". " ) ' 0" "! " ,, " )+ ,' "
( "0 $+ % $ ".
57.' 6 ' +( 7 $ +( 5 ' ). 1 ' + + $' 1 )+ 3" 4 5 '. " ) ' 0" "! " ,, ":
a)) )+ "+ 5 '+ $ +;
b)) )+ "+ 5 '+ % , *) ";
c)' % )+ "+( 5 ' ) " 0 " ! 2 $ +( 5 ';
d)' % )+ "+( 5 ' ) " 0 $ %) ( $ +(;
e)' % )+ "+( 5 ' ) " 0 ( "0 $+ % $ + 5 '.
58.04 ' % 100 % " " ! % $' ) 0. 1 04 )+ 3" , % 10 % " . " ) ' 0" "! " ,, " ' % )+ "+( % " 2 " 0 $' ) 0.
59*. 0 &' ) % 0 ' ) ) 0 16 ) $ ! +( % ' 1$ - "+ & 2'$3 %) & %,' &&+ (& ) ! % ) 2%). -
29
" ) ' 0" "! " ,, " %) $ ! + %+ 2 " 0 ) ' 1 +( & %,' && (.
60*. & '" 100 1% , 1 " '+( 4 – $' ) +. '" 0 &' 1) ! ' 1% %) ' ) + ", " '+ "&' ) + %) & "'$" 0. : ) ) ' 0" "! " ,, " $' ) + 1- % 0 % " " 0 $ & "'$" 0 & ' )?
61.$' 0 ' " , $" 1$+ & % "' * '+ , & 0 5!, " -" * '+ ' 1 +, $' ( %. " ) '- 0" "! " ,, " ' $' &' ) !.
62.1 &0" '" $) , , , , % % 1 %' , )+$' 3" 0 "' ' & , 3" 0 ) '0% ) & '0% & 0) 0. : ) ) ' 0" "!, " & " 0 ) « »?
63.'" ( & " + * '+ " 1 % 7. " ) ' 0"-
"! " ,, " "' % )10"+ & " ) + ) '0% '"
" )0" 357.
64. ' & % 5 ' ' ,, ' 2 ) ,, 2 " ,, 1 - ,, , $,, , " ) , *) " ). 1 ' + & ' %
, % )+ 3" 3 5 '. " ) ' 0" "! " ,, " & 0)0" 0 ) % 34 & '0%: ' +, 2 "+, 1 +.
1.4.3. $ * 2
0" «, "' ) ' 0" "» 0) 0 " 0 $ $4 & 0" 0 « ) ' 0" "» &+" -
' ) ) 1 2 +( ( % ).
"! ) $ "! G ("' 1, $ "! & ", $ "!
&' "' ")) $' " 0 % ". " 1 ", " $' 5- 0 " 2 " « » & & "! ) 3$3 " $ " G. ' -" ) ' 0" "! & & % 0 " ) 3- $ "! g $ - " G &' & '* ! -" " (% , & 4 %, $<)
1 ) " " ' & 2 0 ) $ " G, " ' +
('. 2.1).
G g
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