1. A teacher wants to assign 4 different tasks to her 4 students. In how many ways possible ways can she do it?

2. In a certain general assembly, three major prizes are at state. In how many ways can the first, second, and third prizes can be drawn from a box containing 120 names?

3. In how many different ways can 5 bicycles be parked if there are 7 available parking spaces?

4. How many distinguishable permutations are possible with all the letters of the word ELLIPSES?

5. 8 basketball teams are competing for the top 4 standing to move up to the semifinals. Find the number of possible rankings of the four top teams.

6. In how many different ways can 13 people occupy the 12 seats in the front row of a mini theatre?

7. Find the number of different ways that a family of 6 can be seated around a circular table with 6 chairs.

8. How many 4-digit numbers can be formed from the digits 1,3,5,6,8 and 9 if no repetition is allowed?

9. If there are 10 people and only 6 chairs are available, in how many ways can they be seated?

10. Find the number of distinguishable permutations of the digits of the number 348-838.

With solutions

COMBINATION:
1. You were tasked to take charge of the auditions for the female parts of a stage play. In how many possible ways can you form your cast of 5 female members if there were 15 hopeful?

2. If ice cream is served in a cone, how many ways can Abby choose her three-flavor ice cream scoop if there are 6 available flavors?

3. If each ATM card of a certain bank has to have 4 different digits in its passcode, how many different possibilities of passcode can there be?

4. How many possible permutations are there in three letters of the word PHILIPPINES

5. How many ways can one be seated around a table if 2 insist on sitting beside each other?

With solutions (please)

Since each task is different and each student is assigned only one task, we can use the permutation formula to calculate the number of ways to assign the tasks:

\[ \text{Number of ways} = n! \]

Where \( n \) is the number of tasks (4 in this case) and \( ! \) denotes factorial.

So, for 4 tasks, the number of ways to assign them to 4 students is:

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Therefore, there are 24 possible ways for the teacher to assign 4 different tasks to her 4 students.

To calculate the number of ways to draw the first, second, and third prizes from a box containing 120 names, we can use the concept of permutations.

For the first prize, there are 120 names to choose from. Once the first name is drawn, there are 119 remaining names for the second prize, and once the first two names are drawn, there are 118 remaining names for the third prize.

Therefore, the number of ways to draw the first, second, and third prizes is given by the permutation formula:

\[ \text{Number of ways} = 120 \times 119 \times 118 \]

\[ = \frac{120!}{(120 - 3)!} \]

\[ = \frac{120 \times 119 \times 118}{3 \times 2 \times 1} \]

\[ = 120 \times 119 \times 118 \]

\[ = 168,168,0 \]

So, there are 168,168,0 possible ways to draw the first, second, and third prizes from a box containing 120 names.