Suppose you insert 11 elements into a hash table with 20 bins.  True or False: there must exist **exactly** 9 empty bins, no more, no less, since the number of elements being hashed is less than the number of bins in the hash table by nine. true or false

The statement is **False**.

Here's why:

When you insert 11 elements into a hash table with 20 bins, it is true that you have fewer elements than bins, specifically 9 fewer. However, this does not guarantee that exactly 9 bins will be empty. The distribution of elements into bins depends on the hash function.

### Possible Scenarios:
- **No Collisions**: If the hash function distributes the 11 elements perfectly (with no collisions), then 11 different bins will be occupied, and exactly 9 bins will be empty.
- **Collisions Occur**: If some of the elements hash to the same bin (i.e., collisions occur), fewer than 11 bins may be occupied. This means more than 9 bins could be empty.

Because collisions can cause some bins to be used multiple times, it's possible to have more than 9 empty bins. Therefore, you cannot guarantee that there will be exactly 9 empty bins.

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Would you like more details or have any questions?

Here are 5 related questions:
1. How does the load factor of a hash table relate to the likelihood of collisions?
2. What is the expected number of empty bins if 11 elements are inserted randomly into 20 bins?
3. How does the choice of a hash function impact the distribution of elements in the bins?
4. What is the impact of having too many empty bins in a hash table?
5. How would the time complexity change if the number of elements inserted was equal to the number of bins?

**Tip:** Understanding the distribution properties of your hash function can help in predicting the performance of your hash table and in minimizing collisions.

Explore the distribution of elements in a hash table when inserting 11 elements into 20 bins. Discover why the statement 'There must exist exactly 9 empty bins' is false, with detailed scenarios including collisions and hash function impact.

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