Our problem now is: how many different keys are possible for given. N


How many possible values can a key have?

The number of possible values for a key is simply the total number of values that the key can have. So our one-bit long key can only have two possible values – 0 and 1. If we choose to have a two-bit key it could have one of four possible values – 00, 01, 10 and 11.

How many values can a 3 bit key have?

In fact every time we increase the length of the key by one bit we double the number of possible keys – so a three-bit key has eight possible values – 000, 001, 010, 011, 100, 101, 110 and 111. The total number of keys can be written in scientific form as 2 key length; so a key with a length of eight has 2 8 – that is 256 – values.

How do you divide a problem into three subproblems?

Another approach is to dividethe problem into three subproblems, depending on whether the number of di?erentnumbers showing is 3, 2, or 1. We can count the number of outcomes with three di?erent numbers using thetechnique of Section 4.5. That is, we must select 3 out of the 6possible numberson a die, and we can do so in63= 20 di?erent ways.

What are two paths with no consistent order?

Thetwo paths with noconsistent order never occur. Equation (4.2) tells us that the number of teststmust be at least log23!, which is log26. Since 6 is between 22and 23, we know thatlog26 will be between 2 and 3. Thus, at least some sequences of outcomes in anyalgorithm that sorts three elements must make 3 tests.

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