Counting Problem Discrete Math

I am struggling on this question.

DNA = Four Nucleotides (C,G,A,T)
Cytosine
Guanine
Adenine
Thymine

How many DNA strings of length 6 have exactly two (T) thymine nucleotides?

Ex: CTATAG,GGGTTG,TAAATG not CAAATG nor CTGTTG?

I am not sure where do i go about solving the problem please help. Thanks!

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2

 

First fix the two thymines in the string. This can be done by choosing 22 positions out of 66. Then, in the rest of the positions, fill anything but thymine. This is done in three ways for each of four slots. Can you figure it out now?
– ذ°رپر‚ذ¾ذ½ ذ²ر–ذ»ذ»ذ° ذ¾ذ»ذ¾ر„ ذ¼رچذ»ذ»ذ±رچر€ذ³
2 days ago

  

 

@ذ°رپر‚ذ¾ذ½ذ²ر–ذ»ذ»ذ°ذ¾ذ»ذ¾ر„ذ¼رچذ»ذ»ذ±رچر€ذ³: You should write that as an answer.
– copper.hat
2 days ago

  

 

Does it mean that the answer is 1215 ways?
– Sudeep Baniya
2 days ago

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2 Answers
2

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Choose 22 out of 66 places for the Ts: (62)=15\binom62=15.

Fill the remaining 44 places with any of the other 33 letters: 34=813^4=81.

The answer is therefore 15⋅81=121515\cdot81=1215.

First, fix the two positions in the string where you want to put thymines. This is done in (62)=15\binom 62 = 15 ways.

Next, in the remaining four positions any one of adenine, cytosine and guanine can come. Since these are independent of each other, the number of ways to make this happen is 34=813 ^ 4=81 (4 slots, 3 choices for each slot).

So the answer is 81∗15=121581*15 = 1215 ways.