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:
[tex]_{n}P_{r}\\[/tex]
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 [tex]\frac{n!}{(n-r)!}[/tex]. 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.
[tex]_{5}P_{4}=\frac{5!}{(5-4)!}= \frac{120}{1} =120[/tex]
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 [tex]\frac {n!}{r!(n-r)!}[/tex]. Let's plug in our numbers and solve for this.
[tex]_{8}C_{2}=\frac{8!}{2!(8-2)!} =\frac{8!}{2!(6!)}=\frac{40320}{1440} =28[/tex]
QUESTION 102
Quick note: 0! is equal to 1. I don't really know why but there's probably proof somewhere.
[tex]_{7}C_{0}=\frac{7!}{0!(7-0)!} =\frac{7!}{7!} =1[/tex]
And I believe you already have Question 103 figured out.
4!=4×3×2×1=24
Have a wonderful day! :D
