Hey there! I'm happy to help!
QUESTION 100
Here, we are looking for the number of permutations. A permutation is the number of possible arrangements you can make from a given set of things (order matters). If you have 6 kids standing in alphabetical order in line, that is one permutation. If you do reverse alphabetical order, that is a different permutation.
We are asked to find the number of permutations using the given format:
![_{n}P_{r}\\](https://tex.z-dn.net/?f=_%7Bn%7DP_%7Br%7D%5C%5C)
This is simply a representation of what we want to find. The P means we are looking for the number of permutations. The n is the total number of objects in the set (for example, if there were 26 kids we could choose from to make this 6 person line, n would be 26). The r is the number of things chosen from the set (in our line example, the r would be six because we are choosing 6 kids).
The formula for finding the number of permutations is
. The ! is factorial, it means you multiply that number by every integer counting backwards towards 1. 6! is 6×5×4×3×2×1. There is an option to do this on your calculator, though.
Let's solve this problem.
![_{5}P_{4}=\frac{5!}{(5-4)!}= \frac{120}{1} =120](https://tex.z-dn.net/?f=_%7B5%7DP_%7B4%7D%3D%5Cfrac%7B5%21%7D%7B%285-4%29%21%7D%3D%20%5Cfrac%7B120%7D%7B1%7D%20%3D120)
This means that there are 120 permutations.
QUESTION 101
Now we have a C. This is similar to permutations but with combinations order does not matter. With our line example, having the same six kids in alphabetical order is a different permutation than having it in backwards alphabetical order. With combinations, if you have those specific six kids, it is just one combination; the order does not matter.
The formula for combinations is
. Let's plug in our numbers and solve for this.
![_{8}C_{2}=\frac{8!}{2!(8-2)!} =\frac{8!}{2!(6!)}=\frac{40320}{1440} =28](https://tex.z-dn.net/?f=_%7B8%7DC_%7B2%7D%3D%5Cfrac%7B8%21%7D%7B2%21%288-2%29%21%7D%20%3D%5Cfrac%7B8%21%7D%7B2%21%286%21%29%7D%3D%5Cfrac%7B40320%7D%7B1440%7D%20%3D28)
QUESTION 102
Quick note: 0! is equal to 1. I don't really know why but there's probably proof somewhere.
![_{7}C_{0}=\frac{7!}{0!(7-0)!} =\frac{7!}{7!} =1](https://tex.z-dn.net/?f=_%7B7%7DC_%7B0%7D%3D%5Cfrac%7B7%21%7D%7B0%21%287-0%29%21%7D%20%3D%5Cfrac%7B7%21%7D%7B7%21%7D%20%3D1)
And I believe you already have Question 103 figured out.
4!=4×3×2×1=24
Have a wonderful day! :D