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B(n)=2^(n)
A 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) 
2^(n)
(C) 
2^(n+1)
(D) 
4^(n)

B(n)=2n B(n)=2^{n} \newlineA binary code word of length n n is a string of 00 's and 11 's with n n digits. For example, 10011001 is a binary code word of length 44 . The number of binary code words, B(n) B(n) , of length n n , is shown. If the length is increased from n n to n+1 n+1 , how many more binary code words will there be?\newlineChoose 11 answer:\newline(A) 22\newline(B) 2n 2^{n} \newline(C) 2n+1 2^{n+1} \newline(D) 4n 4^{n}

Full solution

Q. B(n)=2n B(n)=2^{n} \newlineA binary code word of length n n is a string of 00 's and 11 's with n n digits. For example, 10011001 is a binary code word of length 44 . The number of binary code words, B(n) B(n) , of length n n , is shown. If the length is increased from n n to n+1 n+1 , how many more binary code words will there be?\newlineChoose 11 answer:\newline(A) 22\newline(B) 2n 2^{n} \newline(C) 2n+1 2^{n+1} \newline(D) 4n 4^{n}
  1. Problem Understanding: Understand the problem.\newlineWe are given a function B(n)=2nB(n) = 2^n, which represents the number of binary code words of length nn. We need to find out how many additional binary code words are possible when the length increases from nn to n+1n+1.
  2. Calculate Binary Code Words of Length nn: Calculate the number of binary code words of length nn. Using the given function, the number of binary code words of length nn is B(n)=2nB(n) = 2^n.
  3. Calculate Binary Code Words of Length n+1n+1: Calculate the number of binary code words of length n+1n+1. Using the same function, the number of binary code words of length n+1n+1 is B(n+1)=2(n+1)B(n+1) = 2^{(n+1)}.
  4. Determine Increase in Binary Code Words: Determine the increase in the number of binary code words.\newlineTo find out how many more binary code words there are when the length increases from nn to n+1n+1, we subtract the number of binary code words of length nn from the number of binary code words of length n+1n+1.\newlineSo, the increase is B(n+1)B(n)=2(n+1)2nB(n+1) - B(n) = 2^{(n+1)} - 2^n.
  5. Simplify Expression for Increase: Simplify the expression for the increase.\newlineWe can factor out 2n2^n from the expression to simplify it:\newlineB(n+1)B(n)=2n×(21)=2n×1=2nB(n+1) - B(n) = 2^n \times (2 - 1) = 2^n \times 1 = 2^n.
  6. Choose Correct Answer: Choose the correct answer.\newlineThe increase in the number of binary code words when the length increases from nn to n+1n+1 is 2n2^n. This corresponds to answer choice (B) 2n2^n.

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