Question:
Let $a_{n}$ denote the number of all $n$ -digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are $0 .$ Let $b_{n}=$ the number of such $n$ -digit integers ending with digit 1 and $c_{n}=$ the number of such $n$ -digit integers ending with digit $0 .$
Which of the following is correct?
Let $a_{n}$ denote the number of all $n$ -digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are $0 .$ Let $b_{n}=$ the number of such $n$ -digit integers ending with digit 1 and $c_{n}=$ the number of such $n$ -digit integers ending with digit $0 .$
Which of the following is correct?
Updated On: Jul 28, 2023
- $a_{17}=a_{16}+a_{15}$
- $c_{17} \neq c_{16}+c_{15}$
- $b_{17} \neq b_{16}+c_{16}$
- $a_{17}=c_{17}+b_{16}$
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The Correct Option is A
Solution and Explanation
As $a_{n}=a_{n-1}+a_{n-2}$
for $n=17$
$\Rightarrow a_{17}=a_{16}+a_{15}$
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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.
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- 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.
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