(1)已知1Cm5-1Cm6=710Cm7,求C8m;(2)解方程Cx2-x16=C165x-5;(3)计算C100+C111+C122+…+C10090.-数学

题目简介

(1)已知1Cm5-1Cm6=710Cm7,求C8m;(2)解方程Cx2-x16=C165x-5;(3)计算C100+C111+C122+…+C10090.-数学

题目详情

(1)已知
1
Cm5
-
1
Cm6
=
7
10
Cm7
,求C8m
(2)解方程C
x2-x16
=C165x-5
(3)计算C100+C111+C122+…+C10090
题型:解答题难度:中档来源:不详

答案

(1)由已知得
m!(5-m)!
5!
-
m!(6-m)!
6!
=
7(7-m)!m!
10•7!

化简得m2-23m+42=0,
解得m=2或21,
但0≤m≤5,故m=2.
Cm8
=
C28
=class="stub"8×7
2×1
=28

(2)原方程可化为x2-x=5x-5或x2-x=16-(5x-5),
即x2-6x+5=0或x2+4x-21=0,
解得x=1或x=5或x=-7或x=3,
经检验x=5或x=-7不合题意,
故原方程的根为x=1或x=3.
(3)原式=(C110+C111)+C122+…+C10090=(C121+C122)+…+C10090
=(C132+C133)+…+C10090=
C90101
=
C11101

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