Formule du binôme de Newton now easy !

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Formule du binôme de Newton now easy !

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          ( a  +  b ) n =  C  0 n a n +  C  1 n a n − 1 b  +  C  2 n a n − 2 b 2 +  ...  +  C  kn a n − k b k +  ...  +  C  nn b n   n \  k                               C  24  = 6   C  36  = 20    n    ( a  +  b )( a  +  b ) ... ( a  +  b ) = ( a  +  b ) n   n    a  +  b    n   n  = 3  ( a  +  b )( a  +  b )( a  +  b ) = ( aa  +  ab  +  ba  +  bb )( a  +  b )=  aaa  +  aab  +  aba  +  abb  +  baa  +  bab  +  bba  +  bbb  a 3 + 3 a 2 b  + 3 ab 2 +  b 3    a    b       k    n    C  kn  =  n ! k !( n − k )!   k    n    k   k e   k    k    A kn  =  n ( n − 1)( n − 2) ... ( n − k  + 1)    k       A 3100  = 100 × 99 × 98 = 970200   100(100 − 1)(100 − 2)    2 = 3 − 1 =  k − 1    k  = 3   A kn   A kn  =  n !( n − k )!  A 3100  =  100!97!  =  100 × 99 × 98 × 97 × ... × 2 × 197 × ... × 2 × 1   k       k ! =  k ( k − 1)( k − 2) ... × 2 × 1      10 × 9 × 8 × ... × 2 × 1 = 10! = 3628800   k    n    k    n   A kn    k   k e   k    k   k !    k   C  kn    k    n         C  kn  =  A kn k !   C  kn  =  n ! k !( n − k )!   C  kn    A kn    k   k   C  410  = 10 × 9 × 8 × 74 × 3 × 2 × 1 (4 facteurssur 4 facteurs )= 10 × 3 × 7 = 210  n \ k                               ( a  +  b ) n   n  ( a  +  b )( a  +  b )( a  +  b ) ..... ( a  +  b )   n    n   a n   b    C  0 n    n   b   a n   C  0 n a n   C  0 n  = 1    n   a n − 1 b    b    C  1 n   n   a n − 1 b    C  1 n a n − 1 b   a n − k b k   C  kn    C  kn    k    n    b   b n   C  nn  = 1    n    n    C  nn b n  ( a  +  b ) n =  C  0 n a n +  C  1 n a n − 1 b  +  C  2 n a n − 2 b 2 +  ...  +  C  nn b n   ( x  + 3) 6 =  x 6 + 6 x 5 . 3 + 15 x 4 . 9 + 20 x 3 . 27 + 15 x 2 . 81 + 6 x. 243 + 729=  x 6 + 18 x 5 + 135 x 4 + 540 x 3 + 1215 x 2 + 1458 x  + 729   cos 5 x    cos 5 x  = ( e ix +  e − ix 2 ) 5 = 132( e 5 ix + 5 e 4 ix e − ix + 10 e 3 ix e − 2 ix + 10 e 2 ix e − 3 ix + 5 e ix e − 4 ix +  e − 5 ix )= 132( e 5 ix + 5 e 3 ix + 10 e ix + 10 e − ix + 5 e − 3 ix +  e − 5 ix )= 132(2cos(5 x ) + 10 cos (3 x ) + 20 cosx )= 116 cos(5 x ) + 516 cos(3 x ) + 58 cos x      sin(4 x )    cos x    sin x   sin(4 x ) =   ( e 4 ix ) =   (cos(4 x ) +  i sin(4 x ) =   (cos x  +  i sin x ) 4 (cos x  +  i sin x ) 4 = cos 4 x  + 4cos 3 x ( i sin x ) + 6cos 2 x ( i sin x ) 2 + 4cos x ( i sin x ) 3 + ( i sin x ) 4 = cos 4 x  + 4 i cos 3 x sin x − 6cos 2 x sin 2 x − 4 i cos x sin 3 x  + sin 4 x   sin(4 x ) = 4cos 3 x sin x − 4cos x sin 3 x = 4cos x sin x (cos 2 x − sin 2 x )= 4cos x sin x (2cos 2 x − 1)   sin(4 x ) =2sin(2 x )(2cos 2 x − 1) = 2sin(2 x )(1 − 2sin 2 x )    2 x       C  kn  +  C  k − 1 n  =  C  kn +1    C  kn  +  C  k − 1 n  =  n ! k !( n − k )! +  n !( k − 1)!( n − k  + 1)!=  n !( k − 1)!( n − k )!  1 k  + 1 n − k  + 1  =  n !( k − 1)!( n − k )! .n − k  + 1 +  kk ( n − k  + 1)= ( n  + 1)! k !( n − k  + 1)!= ( n  + 1)! k !( n  + 1 − k )!=  C  kn +1     E     n − 1   E   =  { x 1 ,x 2 ,...,x n − 1 }   x n  / ∈  E,E   =  { x 1 ,x 2 ,...,x n − 1 ,x n }    n   k    E     C  kn    x n   x n    A ∪{ x n }    A    k − 1    E     C  k − 1 n − 1   x n    A    E     C  kn − 1   C  kn  =  C  kn − 1  +  C  k − 1 n − 1 
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