BrainDen.com - Brain Teasers  # m00li

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1. m00li's post in Basic Binomial Activity was marked as the answer
2. m00li's post in Rip-off was marked as the answer
Nicely done
3. m00li's post in ABCDCBA was marked as the answer
4. m00li's post in Who Dunnit? was marked as the answer
5. m00li's post in Three questions - Lateral quickie was marked as the answer
6. m00li's post in Urning more balls was marked as the answer
7. m00li's post in A Game of Pirates was marked as the answer
8. m00li's post in Wrist Watch was marked as the answer
9. m00li's post in The "aha!" problems 7. Urning the balls was marked as the answer
continuing...

10. m00li's post in Set nibbling was marked as the answer
11. m00li's post in Sticks Kinematics was marked as the answer
12. m00li's post in Mutual relations was marked as the answer
13. m00li's post in The "aha!" problems - 3 minimum areas was marked as the answer
14. m00li's post in Unique Codes was marked as the answer
EDIT: the last term for F(n) should read n-[n/2]C[n/2]f(n-[n/2]) where [k] is the max integer <=k
15. m00li's post in Two Squares was marked as the answer
I couldn't understand the question. every pair of odd squares and every pair of even square should qualify (e.g. 25-9=4*4, or 16-4=4*3)
16. m00li's post in order of the cards was marked as the answer
17. m00li's post in two pawns was marked as the answer
there are 8 pawns, 2 rooks, 2 knights, 2 bishops, 1 king and 1 queen in white. similarly we have 8,2,2,2,1,1 sets of unique pieces in black.
Method 1:
A = number of ways of selecting 5 out of the above
B = number of ways of selecting 5 such that no pawn is selected
C = number of ways of selecting 5 such that only 1 or 0 white pawn is selected
A = coeff of x5 in (1-x9)2(1-x3)6(1+x)4(1-x)-8 = 2540
B = coeff of x5 in (1-x3)6(1+x)4(1-x)-6 = 876
C = coeff of x5 in (1+x)(1-x9)(1-x3)6(1+x)4(1-x)-7=2230

Method 2
A = no. of ways of selecting 5 such that 2 or 3 or 4 or 5 white pawns are selected
B = no. of ways of selecting 5
C = no. of ways of selecting 5 without any pawns

B = (all 5 pieces are similar) + (4 similar and 1 different) + (3 sim, 2 sim) + (3 sim,2 different) + (2 sim, 2 sim, 1 diff) + (2 sim, 3 diff) + (5 diff)
= (2C1) + (2C1*11C1) + (2C1*7C1) + (2C1*11C2) + (8C2*10) + (8C1*11C3) + (12C5) = 2540
C = (2 sim, 2 sim, 1 diff) + (2 sim, 3 diff) + (5 diff)
= (6C2*8C1) + (6C1*9C3) + (10C5) = 876
A = (5 white pawn) + (4 WP, 1 different) + (3 WP, 2 similar pieces) + (3 WP, 2 different) + (2 WP, 2 sim,1 diff) + (2 WP, 3 diff)
= 1 + (11C1) + (7C1 + 11C2) + (7C1*10C1) + (11C3) = 310