Best Answer m00li, 21 April 2014 - 03:58 AM

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

then Answer = (A-C)/(A-B)

A = coeff of x^{5} in (1-x^{9})^{2}(1-x^{3})^{6}(1+x)^{4}(1-x)^{-8 }= 2540

B = coeff of x^{5} in (1-x^{3})^{6}(1+x)^{4}(1-x)^{-6 }= 876

C = coeff of x^{5} in (1+x)(1-x^{9})(1-x^{3})^{6}(1+x)^{4}(1-x)^{-7}=2230

__Answer = 310/1664 = 0.186298__

**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

Answer = A/(B-C)

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)

= (^{2}C_{1}) + (^{2}C_{1}*^{11}C_{1}) + (^{2}C_{1}*^{7}C_{1}) + (^{2}C_{1}*^{11}C_{2}) + (^{8}C_{2}*10) + (^{8}C_{1}*^{11}C_{3}) + (^{12}C_{5}) = 2540

C = (2 sim, 2 sim, 1 diff) + (2 sim, 3 diff) + (5 diff)

= (^{6}C_{2}*^{8}C_{1}) + (^{6}C_{1}*^{9}C_{3}) + (^{10}C_{5) }= 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 + (^{11}C_{1}) + (^{7}C_{1} + ^{11}C_{2}) + (^{7}C_{1}*^{10}C_{1}) + (^{11}C_{3}) = 310

__Answer = 310/1664 = 0.186298__