+23252 My only way is to try to list them; this is nasty!
Case 1: the subset contains the number 6
A) You choose five numbers lower than 6 (1 way)
and five numbers larger than 6 (1 way) for a total of 1 way
B) You choose four numbers lower than 6 (5 ways)
and four numbers larger than 6 (5 ways) for a total of 5 x 5 = 25 ways
C) You choose three numbers lower than 6 (10 ways)
and three numbers larger than 6 (10 ways) for a total of 10 x 10 = 100 ways
D) You choose two numbers lower than 6 (10 ways)
and to numbers larger than 6 (10 ways) for a total of 10 x 10 = 100 ways
E) You choose one number lower than 6 (5 ways)
and one number larger than 6 (5 ways) for a total of 5 x 5 = 26 ways
F) You choose no number lower than 6 (1 way)
and no number larger than 6 (1 way) for a total of 1 x 1 = 1 way
Now, for the really nasty part, you don't choose the number 6 -- when you do this,
the largest number below 6 and the smallest number above 6 must have a mean
of 6.
A) You choose five numbers lower than 6 (1 way)
and five numbers larger than 6 (1 way) for a total of 1 way
B) You choose four numbers lower than 6 and four numbers larger than 6 (10 ways)
C) You choose three numbers lower than 6 and three numbers larger than 6 (46 ways)
D) You choose two numbers lower than 6 and two numbers larger than 6 (30 ways)
E) You choose one number lower than 6 and one number larger than 6 (5 ways)
Hopefully, I counted correctly -- if not, maybe someone can correct me ...
+118687 Hi Geno, let's see if i get the same :)
How many subsets of the set {1,2,...,11} have median 6?
1 2 3 4 5 6 7 8 9 10 11
The 6 is included
1+(5C4)^2+(5C3)^2+(5C2)^2+(5C1)^1+1
= 1+5^2 +10^2 +10^2 + 5^2 +1
= 1+25+100+100+25+1
=252
6 not included but 5 and 7 both included
(4C4)^2+(4C3)^2+(4C2)^2+(4C1)^2+4C0)^2
=1 +4^2 +6^2 +4^2 +1
=1+16+36+16+1
=70
5,6, and 7 not included but 4 and 8 inlcuded
(3C3)^2+(3C2)^3+(3C1)^2+(3C0)^2
=1+9+9+1
=20
4,5,6,7,8 not included but 3 and 9 included
(2C2)^2+(2C1)^2+(2C0)^2
=1+4+1
=6
3,4,5,6,7,8,9 not included but 2 and 10 included
1+1
= 2
2,3,4,5,6,7,8,9,10 not included but 1 and 11 included
=1
252+70+20+6+2+1 = 351
