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HMMT 十一月 2008 · 冲刺赛 · 第 18 题

HMMT November 2008 — Guts Round — Problem 18

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 10 ] Find the coefficient of x in the expansion of
    ∑6
    6 i
    ( x + 1) · xi =0
    1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUNDst

英文原题

[ 10 ] Find the coefficient of x 6 in the expansion of
( x + 1) 6 ·
6 ∑
i =0
x i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 st HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND

解析

英文解析

  1. [ 10 ] Find the coefficient of x in the expansion of
    ∑6
    6 i
    ( x + 1) · xi =0
    6 i 3
    Answer: 64 Each term of ( x + 1) can be multiplied by a unique power x , 0 ≤ i ≤ 6 to get a sixthdegree term. So the answer is the sum of the coefficients of the terms of ( x + 1) , which is the same as 6
    substituting x = 1 into this power to get 2 = 64.6
    1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUNDst