Question:

Let ana_{n} denote the number of all nn -digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0.0 . Let bn=b_{n}= the number of such nn -digit integers ending with digit 1 and cn=c_{n}= the number of such nn -digit integers ending with digit 0.0 . Which of the following is correct?

Updated On: Jul 28, 2023
  • a17=a16+a15a_{17}=a_{16}+a_{15}
  • c17c16+c15c_{17} \neq c_{16}+c_{15}
  • b17b16+c16b_{17} \neq b_{16}+c_{16}
  • a17=c17+b16a_{17}=c_{17}+b_{16}
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The Correct Option is A

Solution and Explanation

As an=an1+an2a_{n}=a_{n-1}+a_{n-2} for n=17n=17 a17=a16+a15\Rightarrow a_{17}=a_{16}+a_{15}
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Concepts Used:

Permutations and Combinations

Permutation:

Permutation is the method or the act of arranging members of a set into an order or a sequence. 

  • In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point. 
  • A permutation is used in many events of daily life. It is used for a list of data where the data order matters.

Combination:

Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.

  • Combination refers to the combination of about n things taken k at a time without any repetition.
  • The combination is used for a group of data where the order of data does not matter.