Solution Found!
Solution: DNA Sequences (See Example) How many
Chapter , Problem 56AYU(choose chapter or problem)
Problem 56AYU
DNA Sequences (See Example) How many distinguishable DNA sequences can be formed using one A, four Cs, three Gs, and four Ts?
Example:
Problem A DNA sequence consists of a series of letters representing a DNA strand that spells out the genetic code. There are four possible letters (A, C, G, and T), each representing a specific nucleotide base in the DNA strand (adenine, cytosine, guanine, and thymine, respectively). How many distinguishable sequences can be formed using two As, two Cs, three Gs, and one T?
Approach Each sequence formed will have eight letters. To construct each sequence, we need to fill in eight positions with the eight letters:
The process of forming a sequence consists of four tasks:
Task 1: Choose the positions for the two As.
Task 2: Choose the positions for the two Cs.
Task 3: Choose the positions for the three Gs.
Task 4: Choose the position for the one T.
Task 1 can be done in 8C2 ways because we are choosing the 2 positions for A, but order does not matter (because we cannot distinguish the two As). This leaves 6 positions to be filled, so task 2 can be done in 6C2 ways. This leaves 4 positions to be filled, so task 3 can be done in 4C3 ways. The last position can be filled in 1C1 way.
Solution By the Multiplication Rule, the number of possible sequences that can be formed is
There are 1680 possible distinguishable sequences that can be formed.
Questions & Answers
QUESTION:
Problem 56AYU
DNA Sequences (See Example) How many distinguishable DNA sequences can be formed using one A, four Cs, three Gs, and four Ts?
Example:
Problem A DNA sequence consists of a series of letters representing a DNA strand that spells out the genetic code. There are four possible letters (A, C, G, and T), each representing a specific nucleotide base in the DNA strand (adenine, cytosine, guanine, and thymine, respectively). How many distinguishable sequences can be formed using two As, two Cs, three Gs, and one T?
Approach Each sequence formed will have eight letters. To construct each sequence, we need to fill in eight positions with the eight letters:
The process of forming a sequence consists of four tasks:
Task 1: Choose the positions for the two As.
Task 2: Choose the positions for the two Cs.
Task 3: Choose the positions for the three Gs.
Task 4: Choose the position for the one T.
Task 1 can be done in 8C2 ways because we are choosing the 2 positions for A, but order does not matter (because we cannot distinguish the two As). This leaves 6 positions to be filled, so task 2 can be done in 6C2 ways. This leaves 4 positions to be filled, so task 3 can be done in 4C3 ways. The last position can be filled in 1C1 way.
Solution By the Multiplication Rule, the number of possible sequences that can be formed is
There are 1680 possible distinguishable sequences that can be formed.
ANSWER:
Problem 56AYU
Answer:
Step1 of 2:
Approach Each sequence formed will have ten letters. To construct each sequence, we need to fill in ten positions with the ten letters:
The process of forming a sequence consists of four tasks:
Task 1: Choose the positions for the one As.
Task 2: Choose the positions for the four Cs.