Binary Combinations

Binary Numbers
A Binary Number is made up of only 0s and 1s.
So each digit has only two possibilities: 0 or 1

Combinations of "On" and "Off"
So, if one digit has only two possible "positions" (like "0" and "1", or "On" and "Off"), how many different combinations could you have with 2 or more binary digits?
For example, how many different combinations could 4 digits have (such as in our 4 different drums example).
Let's write down all possible combinations, starting with 1 digit (you can test it yourself using the switches):
One digit will have 2 positions...


...two digits have 4 combinations...

0 
0 
→ 
00 
1 
→ 
01 
1 
0 
→ 
10 
1 
→ 
11 

...three digits have 8 combinations...

0 
0 
0 
→ 
000 
1 
→ 
001 
1 
0 
→ 
010 
1 
→ 
011 
1 
0 
0 
→ 
100 
1 
→ 
101 
1 
0 
→ 
110 
1 
→ 
111 

... and four digits have 16 combinations.

0 
0 
0 
0 
→ 
0000 
1 
→ 
0001 
1 
0 
→ 
0010 
1 
→ 
0011 
1 
0 
0 
→ 
0100 
1 
→ 
0101 
1 
0 
→ 
0110 
1 
→ 
0111 
1 
0 
0 
0 
→ 
1000 
1 
→ 
1001 
1 
0 
→ 
1010 
1 
→ 
1011 
1 
0 
0 
→ 
1100 
1 
→ 
1101 
1 
0 
→ 
1110 
1 
→ 
1111 

And, in fact, we have created the first 16 binary numbers:
Decimal: 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
Binary: 
0 
1 
10 
11 
100 
101 
110 
111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
This is quite a useful thing to remember. If ever you forget how the sequence of binary numbers go, just think: "0" and "1", then "0" and "1" again but with a "1" in front of each ("10" and "11") then take those four and put "1"s in front ("100","101","110","111") and so on!
Binary Digits Double
Also notice that each time you add another binary digit you double the combinations. Why double? Because you have to take all the previous combinations and match them with a "0" and a "1" like above.
So, if you had 5 things, then the total combinations would be 32, 6 things would be 64, etc.
Using exponents, the combinations can be shown as:
No of Digits 
Formula 
Combinations 
1 
2^{1} 
2 
2 
2^{2} 
4 
3 
2^{3} 
8 
4 
2^{4} 
16 
5 
2^{5} 
32 
6 
2^{6} 
64 
etc... 
etc.^{..} 
etc... 
Example: if you had 50 binary digits (or even 50 things that can only have two positions each), how many combinations would that be?
Answer 2^{50} = 2 × 2 × 2 × 2 ... (fifty of these) = 1,125,899,906,842,624
So, a binary number with 50 digits could have 1,125,899,906,842,624 combinations.
Or to put it another way, it could show a number up to 1,125,899,906,842,623 (note that this is one less than the total combinations, because one of the combinations is 0).

Chess Board
There is an old Indian legend about a King who was challenged to a game of chess by a visiting Sage. The King asked "what will be the prize if you win?".
The Sage said he would simply like some grains of rice: one on the first square, 2 on the second, 4 on the third and so on, doubling on each square. The King was surprised by this humble request.

Well, the Sage won, so how many grains of rice should he receive?
On the first square: 1 grain, on the second square: 2 grains (for a total of 3) and so on like this:
Square 
Grains 
Total 
1 
1^{} 
1 
2 
2^{} 
3 
3 
4^{} 
7 
4 
8^{} 
15 



10 
512^{} 
1,027 



20 
524,288

1,048,575 



30 
53,6870,912

1,073,741,823 



64 
???

??? 
By the 30th square you can see it is already a lot of rice! A billion grains of rice would weigh 25 tonnes (1,000 grains weigh about 25g  I weighed some).
Notice that the Total of any square is 1 less than the Grains on the next square (Example: square 3's total is 7, and square 4 has 8 grains). So the total of all squares is a formula: 2^{n}1, where n is the number of the square. For example, for square 3, the total is 2^{3}1 = 81 = 7
So, to fill all 64 squares in a chess board would need 2^{64}1 = 18,446,744,073,709,551,615 grains (460 billion tonnes of rice), many times more rice than in the whole kingdom.
So, the power of binary doubling is nothing to be taken lightly (460 billion tonnes is not light!)
(By the way, in the legend the Sage reveals himself to be Lord Krishna and tells the King that he doesn't have to pay the debt immediately but can pay him over time, just serve rice to pilgrims every day until the debt is paid off.)
Hexadecimal
Lastly, I would like to tell you about the special relationship between Binary and Hexadecimal.
There are 16 Hexadecimal digits, and we already know that 4 binary digits have 16 combinations. Well, this is exactly how they relate to each other:
Binary: 
0 
1 
10 
11 
100 
101 
110 
111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
Hexadecimal: 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
A 
B 
C 
D 
E 
F 
So, when people use computers (which prefer binary numbers), it is a lot easier to use the single hexadecimal digit rather than 4 binary digits.
For example, the binary number "100110110100" is "9B4" in hexadecimal. I know which I would prefer to write!
