HMMT 十一月 2017 · THM 赛 · 第 6 题

HMMT November 2017 — THM Round — Problem 6

R thea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea? 2 20

解析

R thea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea? Proposed by: Yuan Yao 4 Answer: 12 · 7 or 28812 There are 4 · 3 = 12 ways to choose the first base pairs, and regardless of which base pair it is, there are 3 possibilities for the next base on one strand and 3 possibilities for the next base on the other strand. Among these possibilities, exactly 2 of them have identical bases forming a base pair (using one of the base not in the previous base pair if the previous pair is H-M or N-T, or one of the base in the previous pair otherwise), which is not allowed. Therefore there are 3 · 3 − 2 = 7 ways to choose 4 each of the following base pairs. Thus in total there are 12 · 7 = 28812 possible DNA (which is also the maximum number of species). 2 20