4. Probability
Counting
1:56 minutesProblem 4.4.16
Textbook Question
DNA Nucleotides DNA (deoxyribonucleic acid) is made of nucleotides. Each nucleotide can contain any one of these nitrogenous bases: A (adenine), G (guanine), C (cytosine), T (thymine). If one of those four bases (A, G, C, T) must be selected three times to form a linear triplet, how many different triplets are possible? All four bases can be selected for each of the three components of the triplet.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with determining the number of different triplets that can be formed using the four nitrogenous bases (A, G, C, T), where each base can be selected for each position in the triplet.
Step 2: Recognize that this is a problem involving permutations with repetition. Since there are 4 possible choices (A, G, C, T) for each of the 3 positions in the triplet, the total number of combinations can be calculated using the formula for permutations with repetition: \( n^r \), where \( n \) is the number of choices per position and \( r \) is the number of positions.
Step 3: Substitute the values into the formula. Here, \( n = 4 \) (the four bases: A, G, C, T) and \( r = 3 \) (the three positions in the triplet). The formula becomes \( 4^3 \).
Step 4: Simplify the expression \( 4^3 \) to determine the total number of possible triplets. This involves multiplying 4 by itself three times: \( 4 \times 4 \times 4 \).
Step 5: Conclude that the result represents the total number of unique triplets that can be formed when each of the four bases can be selected for each of the three positions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combinatorial Counting
Combinatorial counting is a fundamental principle in mathematics used to determine the number of ways to arrange or select items. In this context, it involves calculating the total combinations of nucleotide bases that can form a triplet. Since each position in the triplet can independently be any of the four bases, the total number of combinations can be found using the formula for permutations with repetition.
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Permutations with Repetition
Permutations with repetition refer to the arrangement of items where some items can be repeated. In the case of forming a triplet from four bases, each of the three positions in the triplet can be filled by any of the four bases, leading to a scenario where the same base can appear multiple times in different positions. This concept is crucial for calculating the total number of unique triplets possible.
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Exponential Growth in Combinations
Exponential growth in combinations occurs when the number of choices increases exponentially with the number of selections. For this problem, since there are four choices for each of the three positions in the triplet, the total number of combinations is calculated as 4 raised to the power of 3. This illustrates how quickly the number of possible combinations can grow as more positions or choices are added.
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