IIT-JEE 2012 · Paper 2 · Mathematics Q49

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MathematicsSingle correct+3 / -1Question 49 of 60

Passage

Paragraph for Questions 49 and 50

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.

The value of $b_6$ is

Answer and worked solution

Official answer: B

Recurrences. The leading digit is 1. A valid number ending in 1 is any valid $(n-1)$ -digit number followed by 1; one ending in 0 must have 1 before the 0:
$b_n=a_{n-1},\qquad c_n=b_{n-1},\qquad a_n=b_n+c_n.$

Table.

$n$	1	2	3	4	5	6
$b_n$	1	1	2	3	5	8
$c_n$	0	1	1	2	3	5
$a_n$	1	2	3	5	8	13

(B) $b_6=8$

Topics:Fundamental principle of counting

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IIT-JEE 2012 · Paper 2, Mathematics question 49