$36 \cdot 61$ In the final stage of a professional bowling tournament, the top five players compete as follows: first, the fifth place player competes with the fourth place player, the loser gets the 5th prize; the winner competes with the third place player, the loser gets the 4th prize; the winner competes with the second place player, the loser gets the 3rd prize; the winner competes with the first place player, the loser gets the 2nd prize; the winner gets the 1st prize. The number of different prize-winning sequences is
(A) 10.
(B) 16.
(C) 24.
(D) 120.
(E) None of the above answers.
(39th American High School Mathematics Examination, 1988)
[Solution] Let the top five be $A, B, C, D, E$; then the way $D, E$ compete to decide the 5th place is 2 ways; the winner of $D, E$ and $C$ compete to decide the 4th place also has 2 ways; the winner among them and $B$ compete to decide the 3rd place has 2 ways; finally, the winner of the first three rounds and $A$ compete to decide the 2nd and 1st place also has 2 ways. Therefore, the total number of ways is $2 \cdot 2 \cdot 2 \cdot 2=2^{4}=16$ (ways). Hence, the answer is $(B)$.