Solve...... help

Question

1. 8P5
2. 20C17
3. A 3-digit security code can use the numbers 0-9. how many possible combinations are there if the numbers can be repeated?
4. The senior class is electing 3 class officers: president, vice president, and secretary. If there are 36 seniors, how many ways can they be elected?
5. If a bag contains 18 different-colored marbles, how many different samples of 3 marbles can be drawn from the bag?

Answer 1

1, 2 I will address this at the end

3)

Let's think about this. We are combining 3 digits to make a number. So it's like for 3 places there are total 10 candidates (because it's 0 to 9).

So there are 10 options for each spot (because numbers are repeated you can use all the numbers).

Resulting in 10 * 10 * 10 = 1000

4)

This one is different from previous one.

There are total of 36 people as candidate.

But once a position has filled out you have one less candidate.

You end up with 36 * 35 * 34 because one person is drawn from the pool everytime

5)

This one is different in perspective to that there is no specific spot for the room.

For example in problem 3 and 4, the order and the spot that needed to be filled from the candidate was clear. Therefore order or orientation of how the candidates are filled matter. However for this case it doesn't matter because we do not distinguish between blue blue red and red blue blue as we did for earlier ones. (For example for number 4 we would distinguish between Student A as president vs Student A as secretary)

So for this one You need to divide by all the numbers of different ordering which would be 3 * 2 * 1

The answer comes out to be (18 * 17 * 16) / (3 * 2 * 1)

(18 * 17 * 16) comes from the fact that marbles are drawn from the bag each time.

(3 * 2 * 1) comes from all the possible orientation of the ordering.

for 1. (8 * 7 * 6 * 5 * 4) which is a concept of permutation which is used in number 4

also written as 8!/ 5!

for 2. 20! / (17!3!) which is concept of combination used for number 5