In the binary number system which is used in computer operations, there are only two digits allowed: 0 and 1. How many different binary numbers can be formed using at most four binary digits?

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Answer 1 · 2018-05-15T11:58:23

$30$ $30$

The total number is the sum of all possible numbers with one, two, three or four digits.

There are exactly $2^{n}$ $2^n$ numbers with $n$ $n$ digits: for each digit you have two choices (it can be either $0$ $0$ or $1$ $1$ ). So, you have

$2^{1} = 2$ $2^1 = 2$ numbers with one digit ( $0$ $0$ and $1$ $1$ )
$2^{2} = 4$ $2^2 = 4$ numbers with two digits ( $00$ $00$ , $01$ $01$ , $10$ $10$ , $11$ $11$ )
$2^{3} = 8$ $2^3 = 8$ numbers with three digits ( $000$ $000$ , $001$ $001$ , $010$ $010$ , $011$ $011$ , $100$ $100$ , $101$ $101$ , $110$ $110$ , $111$ $111$ )
$2^{4} = 16$ $2^4 = 16$ numbers with three digits ( $0000$ $0000$ , $0001$ $0001$ , $0010$ $0010$ , $0011$ $0011$ , $0100$ $0100$ , $0101$ $0101$ , $0110$ $0110$ , $0111$ $0111$ , $1000$ $1000$ , $1001$ $1001$ , $1010$ $1010$ , $1011$ $1011$ , $1100$ $1100$ , $1101$ $1101$ , $1110$ $1110$ , $1111$ $1111$ )

So, you have $2 + 4 + 8 + 16 = 30$ $2+4+8+16 = 30$ numbers with at most four digits.