Question

1. A standard license plate consists of an ordered string of 3 letters followed by 4 digits. How many license plates can be issued using this format? 2. A DNA sequence is a string of elements chosen among the four possible bases A, C,G, and T. How many 4-element DNA sequences: (a) Do not contain the base T? (b) Contain the sequence ACG? (c) Contain all four bases? (d) Contain at most three of the four bases?

Answer 1

1) 17,576,000

2) (a) The number of 4-element DNA sequences that do not contain the base T is 81.

(b) The number of 4-element DNA sequences that contain the sequence ACG is 4.

(c) The number of 4-element DNA sequences that contain all four bases is 4.

(d) The number of 4-element DNA sequences that contain at most three of the four bases is 255.

Definitions

--> DNA is usually defined as deoxyribonucleic acid: an extremely long macromolecule that is the main component of chromosomes and is the material that transfers genetic characteristic in all life forms, constructed of two nucleotide strands coiled around each other in a ladderlike arrangement with the sidepieces composed of alternating phosphate and deoxyribose units and the rungs composed of the purine and pyrimidine bases adenine, guanine, cytosine, and thymine: the genetic information of DNA is encoded in the sequence of the bases and is transcribed as the strands unwind and replicate.

---> Adenine is a purine base, C5H5N5, one of the fundamental components of nucleic acids, as DNA, in which it forms a base pair with thymine, and RNA in which it pairs with uracil. Symbol: A

---> Cytosine is a pyrimidine base, C4H5N3O, that is one of the fundamental components of DNA and RNA, in which it forms a base pair with guanine. Symbol: C.

---> Guanine is a purine base, C5H5N5O, that is a fundamental constituent of DNA and RNA, in which it forms base pairs with cytosine. Symbol: G.

---> Thymine is a pyrimidine base, C5H6N2O2, that is one of the principal components of DNA, in which it is paired with adenine. Symbol: T.

1) For the first position, we have 26 choices (A to Z) for the letter.

For the second position, we also have 26 choices.

For the third position, we have 26 choices.

For the fourth position, we have 10 choices (0 to 9) for the digit.

For the fifth position, we have 10 choices.

For the sixth position, we have 10 choices.

For the seventh position, we have 10 choices.

Therefore, the total number of possible license plates is:

26 × 26 × 26 × 10 × 10 × 10 × 10 = 17,576,000

2) (a) To count the number of 4-element DNA sequences that do not contain the base T, we have three choices for each position (A, C, or G). Therefore, the total number of such sequences is:

3 × 3 × 3 × 3 = 81

(b) To count the number of 4-element DNA sequences that contain the sequence ACG, we can fix the first three positions to A, C, and G, respectively, and then have four choices for the fourth position (A, C, G, or T). Therefore, the total number of such sequences is: 4

(c) To count the number of 4-element DNA sequences that contain all four bases, we can fix the first three positions to A, C, and G, respectively, and then have four choices for the fourth position (since we need to include T). Therefore, the total number of such sequences is: 4

(d) To count the number of 4-element DNA sequences that contain at most three of the four bases, we can count the total number of 4-element DNA sequences and subtract the number of sequences that contain all four bases. The total number of 4-element DNA sequences is:

4 × 4 × 4 × 4 = 256

The number of sequences that contain all four bases is:

1 (the sequence ACGT)

Therefore, the number of sequences that contain at most three of the four bases is:

256 - 1 = 255

So there are 81 4-element DNA sequences that do not contain the base T, 4 that contain the sequence ACG, 4 that contain all four bases, and 255 that contain at most three of the four bases.

Answer 2

1.) To calculate the number of license plates that can be issued using the given format, we need to determine the number of possibilities for each position in the string.

For the letters, there are 26 possible choices (A-Z) for each of the three positions. Therefore, the number of possibilities for the letters is 26^3.

For the digits, there are 10 possible choices (0-9) for each of the four positions. Therefore, the number of possibilities for the digits is 10^4.

To find the total number of license plates, we multiply the number of possibilities for the letters by the number of possibilities for the digits:

Total number of license plates = 26^3 * 10^4

2.)

(a) To find the number of 4-element DNA sequences that do not contain the base T, we need to consider the three remaining bases (A, C, G). For each position, we have three choices. Therefore, the total number of 4-element DNA sequences without T is 3^4.

(b) To find the number of 4-element DNA sequences that contain the sequence ACG, we fix the positions of A, C, and G. The remaining position can have any of the four bases (A, C, G, T). Therefore, the total number of 4-element DNA sequences containing ACG is 1 * 1 * 1 * 4 = 4.

(c) To find the number of 4-element DNA sequences that contain all four bases, we fix the positions of A, C, G, and T. There is only one possibility for each position. Therefore, the total number of 4-element DNA sequences containing all four bases is 1.

(d) To find the number of 4-element DNA sequences that contain at most three of the four bases, we subtract the number of sequences that contain all four bases from the total number of possible sequences. Therefore, the total number of 4-element DNA sequences containing at most three of the four bases is 4^4 - 1.