List all the sequences of zeros and ones of length 2 that do

List all the sequences of zeros and ones, of length 2, that do not have both first and last element a one. List all the sequences of zeros and ones, of length 4, that do not have both first and last element a one. Suppose n Greaterthanorequalto 2. How many sequences of zeros and ones, of length n, are there that do not have both first and last element a one.

Solution

a. 00, 01, 10 So total 3 ways. Beacuse both 1st and last elemnet should not be 1.

b. Total cases are

0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111

Therefore total no. of cases with 4 digit is 2^4= 16

Now reduce the cases where 1 ia at both 1st and 4th place

They are 1001, 1011, 1101, 1111 Total 4 cases

Therefore desired list contain only 16 - 4 = 12 cases

So eliminate these above 4 cases from 16 cases to get the answer.

c. n >= 2

Total possible way to arrange without any restriction = 2^n

Now eliminate the 1st and last position, so total no of ways = 2^(n-2)

Now fill the 1st and last place so that at both place 1 should not come

So the possible pair will be 00, 01 & 10, Total 3 ways

Therefore total possiblw ways = 3 x 2^(n-2)