Let an denote the number of all n-digit positive integers formed by the digit 0,1 or both such that no - Maths - Permutations and Combinations

Question

Let an denote the number of all n-digit positive
integers formed by the digit 0,1 or both such that no consecutive
digit 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
I Which of the following is correct of:
(a) a17 = a16 + a15 (b)
c17 is not = c16 + c15
(c) b17 is not = b16 + c16 (d)
a17 = c17 + b16
II The value of b6 is :
(a) 7 (b) 8 (c) 9 (d) 11

Anuradha Sharma · Accepted Answer

Note that a1=1 and a2=2For n≥3, if the last digit is 1, then first n-1 digit can be choosen in an-1ways and if the last digit is 0 , then n-1th digit is 1 and the first n-1digits can be chosen in an-2 ways, Thusan=an-1+an-2 ∀ n≥3
1Also note that bn=an-1 ∀ n≥3and cn=an-2 ∀ n≥3Now we have, b6=a5=a4+a3=a3+a2+a3=2a2+a1+a2=32+21=8And putting n = 17 in equation1 we get, a17=a16+a15