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Question

Let an denote the number of all n digit positive integers formed by the digits 0,1, of both such that no consecutive digits in them are 0. Let bn= the number of such n digit integers ending with digit 1 and cn= the number of such n digit integers ending with digit 0. The value of b6 is


A

7

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B

8

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C

9

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D

11

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Solution

The correct option is B

8


Explanation for the correct option:

Finding the value of b6:

Given an denote the number of all n digit positive integers formed by the digits 0,1,

Therefor,

For a1total number =1 since we have only '1' as a positive integer formed by the digits 0,1,

For a2total numbers =2 As we have numbers 10,11

For a3total numbers =3 As we have numbers 101,110,111

According to given information we will not take 100 in above set because two consecutive digits are zero.

For a4total numbers =5 As we have numbers 1010,1011,1101,1111,1110

Observing a1, a2, a3,a4 we get a relation

an=an-1+an-2

Also, given bn= the number of such n digit integers ending with digit 1

Similarly

b1=1b2=11b3=101,111b4=1011,1101,1111

Similarly we get the relation,

bn=bn-1+bn-2,n>3

Substituting n=6

b6=b6-1+b6-2=b5+b4=b4+b3+b4[Againbythisrelation]=2×(numberofelementinb4)+(numberofelementinb3)=2×3+2b6=8

Hence, option (B) is the correct answer.


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