a binary digit or"bit" is either 0 or 1. a nybble is a four bit sequence. how many different nybbles are possible?

Question

Combinations I need help with this math problem!!!

Answer 1

Recall the product rule of combinatorics:

If you must make n independent decisions, and the numbers of choices for those decisions are k₁, ... ,k_n respectively, then the number of ways for me to make the decisions is k₁*...*k_n

If you have the same number of choices for all these decisions, say k choices for each, then the number of ways of making all decisions is kⁿ

In your problem, you have four independent decisions, one for each bit. You have two choices for each, namely, a 0 or 1. Thus the number of ways is 2⁴ = 16. Clearly each of these 16 possibilities yield a different possibility so there are 16 nybbles.

Answer 2

Hi Kim,

Since there aren't any restrictions placed on the nybbles each bit is not dependent on any other bit this is important because it means that:

• The first digit in the nybble (the first bit) has two options -- either a 0 or 1
• The second digit in the nybble (the second bit) also has only two options -- either a 0 or 1 -- regardless of the value of the first bit!

If we stopped there there would be 2 * 2 = 4 possibilities since each possible value for the first bit gets paired with two options in the second bit.  (With only 4 possibilities we can even list them out fairly easily to make sure we are correct -- 00, 01, 10, 11 are the only options.)

Continuing on the third and fourth bits only have 2 options also so overall there are 2*2*2*2 = 2^4 = 16 nybbles possible.

(Note: In general if you have n repetitions of the same event, each that has k choices, then there are k^n possible outcomes.  Here you had n=4 repetitions (the four bits making up the nybble) and k=2 choices per bit so there were 2^4 outcomes!)