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The opinions expressed herein are my own personal opinions and do not represent anyone else's view in any way, including those of my employer.
© Copyright 2013
The opinions expressed herein are my own personal opinions and do not represent anyone else's view in any way, including those of my employer.
© Copyright 2013
(1) 求 (1 + x + x2 + x 3+ x4 + x5 + x6 )3 展式中， x10 的係數
(2) 求 (1 + x + x2 + x3 + x4 ) 5展式中， x3 的係數
(1)
(1 + x + x2 + x 3+ x4 + x5 + x6 )3
=(1 + x + x2 + x 3+ x4 + x5 + x6 ) (1 + x + x2 + x 3+ x4 + x5 + x6 ) (1 + x + x2 + x 3+ x4 +x5 + x6 )
其展式的每一項，是從每一個()之中，恰取一項出來相乘而得，
今要求 x10的係數，可能是(6,4,0)，
即第一個()之中，取 x6
而第二個()之中，取 x4
第三個()之中，取 x0(就是1)
也可能是 (6,0,4), (4,0,6), (4,6,0), (0,4,6), (0,6,4),
這種情況共得到 6種，依此， x10的係數來源如列表:
|
(6,4,0) |
6 |
|
(6,3,1) |
6 |
|
(6,2,2) |
3 |
|
(5,5,0) |
3 |
|
(5,4,1) |
6 |
|
(5,3,2) |
6 |
|
(4,4,2) |
3 |
|
(4,3,3) |
3 |
綜上所述，答=36
(2)同理 (1 + x + x2 + x3 + x4 )5
等於5個 (1 + x + x2 + x3 + x4 )相乘，每個()恰取一項:
(1)1個()取x3,4個()取1，方法5
(2)1個()取x2, 1個()取x，3個()取1，方法20
(3)3個()取x,2個()取1，方法10
答:35
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