B(n)=2nA binary code word of length n is a string of 0 's and 1 's with n digits. For example, 1001 is a binary code word of length 4 . The number of binary code words, B(n), of length n, is shown. If the length is increased from n to n+1, how many more binary code words will there be?Choose 1 answer:(A) 2(B) 2n(C) 2n+1(D) 4n
Q. B(n)=2nA binary code word of length n is a string of 0 's and 1 's with n digits. For example, 1001 is a binary code word of length 4 . The number of binary code words, B(n), of length n, is shown. If the length is increased from n to n+1, how many more binary code words will there be?Choose 1 answer:(A) 2(B) 2n(C) 2n+1(D) 4n
Problem Understanding: Understand the problem.We are given a function B(n)=2n, which represents the number of binary code words of length n. We need to find out how many additional binary code words are possible when the length increases from n to n+1.
Calculate Binary Code Words of Length n: Calculate the number of binary code words of length n. Using the given function, the number of binary code words of length n is B(n)=2n.
Calculate Binary Code Words of Length n+1: Calculate the number of binary code words of length n+1. Using the same function, the number of binary code words of length n+1 is B(n+1)=2(n+1).
Determine Increase in Binary Code Words: Determine the increase in the number of binary code words.To find out how many more binary code words there are when the length increases from n to n+1, we subtract the number of binary code words of length n from the number of binary code words of length n+1.So, the increase is B(n+1)−B(n)=2(n+1)−2n.
Simplify Expression for Increase: Simplify the expression for the increase.We can factor out 2n from the expression to simplify it:B(n+1)−B(n)=2n×(2−1)=2n×1=2n.
Choose Correct Answer: Choose the correct answer.The increase in the number of binary code words when the length increases from n to n+1 is 2n. This corresponds to answer choice (B) 2n.
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