问题1.评估以下内容:
i) 14碳3
解决方案:
We know that nCr=n!/(n-r)!r!
=>14C3=14!/(14-3)!3!
=14!/11!3!
=14x13x12/3x2x1
=364
ii) 12碳10
解决方案:
= 12!/(12-10)!10!
= 12!/2!10!
= 12×11/2×1
= 66
iii) 35 ℃ 35
解决方案:
= 35!/(35-35)!35!
= 1
iv) n + 1 C n
解决方案:
= (n+1)!/(n+1-n)!n!
= (n+1)!/n!
= n+1
v)5
解决方案:
∑ 5Cr=5C1+5C2+5C3+5C4+5C5
r = 1
= 5+10+10+5+1
= 31
问题2.如果n C 12 = n C 5 ,则找到n的值。
解决方案:
Given that nC12=nC5.
We know that two combinations will be equal when the sum of their r’s is equal to n.
=>n=12+5=17.
问题3.如果n C 4 = n C 6 ,则找到12 C n 。
解决方案:
=>n=6+4=10
=>12C10=12!/10!2!
=12×11/2
=66
问题4.如果n C 10 = n C 12 ,则为23 C n 。
解决方案:
n = 10+12=22
=>23C22 = 23!/22!1!
= 23
问题5.如果24 C x = 24 C 2x + 3 ,则找到x。
解决方案:
24 = x+2x+3
24 = 3x+3
21 = 3x
x = 21/3
x = 7
问题6.如果18 C x = 18 C x + 2 ,则找到x。
解决方案:
18 = x+x+2
18 = 2x+2
16 = 2x
x = 8
问题7.如果15 C 3r = 15 C r + 3 ,则找到r。
解决方案:
15 = 3r+r+3
15 = 4r+3
12 = 4r
r = 3
问题8.如果8 C r – 7 C 3 = 7 C 2 ,则找到r。
解决方案:
Given 8Cr–7C3=7C2
=>8Cr=7C2+7C3
We know that nCr+nCr-1=n+1Cr
=>8Cr=8C3
=>r=3
问题9.如果15 C r : 15 C r-1 = 11:5,则找到r。
解决方案:
15Cr/15Cr-1=11/5
(15!/(15-r)!r!)/(15!/(15-r+1)!(r-1)!)=11/5
15-r+1/r = 11/5
5(16-r) = 11r
80-5r = 11r
16r = 80
r = 5
问题10.如果n + 2 C 8 : n-2 P 4 = 57:16,则找到n。
解决方案:
We know that nPr=n!/(n-r)!
=>((n+2)!/(n+2-8)!8!)/((n-2)!/(n-2-4)!)=57/16
=>(n+2)(n+1)(n)(n-1)/8!=57/16
=>(n-1)n(n+1)(n+2)=(57/16)8!
=>(n-1)n(n+1)(n+2)=57×7!/2
=>(n-1)n(n+1)(n+2)=57x7x6x5x4x3
=>(n-1)n(n+1)(n+2)=19x3x7x6x5x4x3
=>(n-1)n(n+1)(n+2)=19x18x20x21
=>n=19