Consider all the strings of n -digits which can be formed using the numbers from the set 0,1,2,3 (may be starting with zero also). If E _ n denotes the number of strings containing even number of zeros and O _n denotes the strings containin

Question

Consider all the strings of n -digits which can be formed using the numbers from the set 0,1,2,3 (may be starting with zero also). If E _ n denotes the number of strings containing even number of zeros and O _n denotes the strings containing odd number of zeros then

Quantrex Academy · Accepted Answer

Correct answer: C. O _ n = 4^ n -2^ n 2

aligned & E _n=3^n+ ^n C₂ 3^ n-2 + ^n C₄ 3^ n-4 + & O _n= ^n C₁ 3^ n-1 + ^n C₃ 3^ n-3 + aligned Adding and subtracting, we get E_n+O_n=(3+1)^n=4^n, E_n-O_n=(3-1)^2=2^n Adding and subtracting again, we get 2 E_n=4^n+2^n E_n= 4^n+2^n 2 2 O _n=4^n-2^n O_n= 4^n-2^n 2

Consider all the strings of n -digits which can be formed using the numbers from the set 0,1,2,3 (may be starting with zero also). If E _ n denotes the number of strings containing even number of zeros and O _n denotes the strings containing odd number of zeros then

Options

Correct answer

Step-by-step solution

More from Permutation and Combination