I have no idea how to answer these (note: there are just a couple questions out of 52, and I can't for the life of me answer them).

1. In how many ways can you have a snack if you find 3 apples, 2 bananas and 4 cookies on the kitchen counter?

2. Solve the following equation for n, where n is a natural number.
C (n+1, 3) = C(n,2)

3. A box contains 6 blue socks and 4 white socks. Find the number of ways 2 socks can be drawn from the box where:
1. there are no restrictions
2. they are different colors
3. they are the same color
4. A teacher organizing a field trip finds that 50 students have signed up. However, the bus has only 47 seats, so a few students will have to travel by car. The teacher and one other supervisor must go on the bus. Explain two different methods for using combinations to find how each teacher can choose which students go on the bus. Show that both methods produce the same answer.

2. Hello, acc!

There is a puzzling word in this problem . . .

4. A teacher organizing a field trip finds that 50 students have signed up.
However, the bus has only 47 seats, so a few students will have to travel by car.
The teacher and one other supervisor must go on the bus.

Explain two different methods for using combinations to find
how each teacher can choose which students go on the bus.

Do both teachers participate in the decisions? .If so, how?

Show that both methods produce the same answer.
I will ignore the word "each."

Two teachers and 45 students will ride on the bus.
The other five students will travel by car.

[1] Choose the 45 students to ride on the bus: .$\displaystyle _{50}C_{45} \:=\:\frac{50!}{45!\,5!}$ ways.

[2] Choose the 5 students to ride in the car: .$\displaystyle _{50}C_5 \:=\:\frac{50!}{5!\,45!}$ ways.

3. Originally Posted by acc
I have no idea how to answer these (note: there are just a couple questions out of 52, and I can't for the life of me answer them).

1. In how many ways can you have a snack if you find 3 apples, 2 bananas and 4 cookies on the kitchen counter?

2. Solve the following equation for n, where n is a natural number.
C (n+1, 3) = C(n,2)

3. A box contains 6 blue socks and 4 white socks. Find the number of ways 2 socks can be drawn from the box where:
1. there are no restrictions
2. they are different colors
3. they are the same color

4. A teacher organizing a field trip finds that 50 students have signed up. However, the bus has only 47 seats, so a few students will have to travel by car. The teacher and one other supervisor must go on the bus. Explain two different methods for using combinations to find how each teacher can choose which students go on the bus. Show that both methods produce the same answer.
Well, I'll do some of it, but not all. Once you understand basic concepts, you should be able to apply them to more complex situations.

1. Assume the apples are identical, the bananas are identical, etc. Then you can either choose 0, 1, 2, or 3 apples, and you can choose either 0, 1, or 2 bananas, and similarly with cookies. So you have 4*3*5 = 60 ways to choose your meal.

This is a basic application of the counting principle, which hopefully you understand. I'll copy and paste a basic example I wrote in another post not long ago.

Here's how the principle works. Say you're building a house, and you have two types of roof to choose from, and three types of windows, and that's all you're concerned about. So the number of ways to choose roof and windows is 2*3 = 6.

You can list them to see why. Say roofs are either A = angled, F = flat. Windows are either O = opaque, T = translucent, C = clear. (Yes, opaque windows are silly, but this is just an example.) Then the 6 possibilities are:

A, O
A, T
A, C
F, O
F, T
F, C

Make sense?

2. Do you know how to write C(n, k) with factorials?

3.1. We assume that the socks are non-identical, because otherwise the problem would be way too simple.

Here we simply need to choose 2 items out of 10. The answer is C(10, 2).

3.2. There are C(6, 1) = 6 ways to choose a blue sock and 4 ways to choose a white sock. So the answer is 6*4 = 24.

3.3. You can either get two blue socks or two white socks. For blue, you have C(6,2). For white, you have C(4,2). You need to add these together to get C(6,2) + C(4,2). You are adding because these are separate events; they cannot happen simultaneously. Try listing them out on paper to see why we add instead of multiply.

4. It seems we're supposed to assume that we want as many students on the bus as possible, so we have 45 students chosen to be on a bus out of 50 students total. This is simply C(50, 45).

I'm not sure what the problem designer intended as a second way to count them. My guess is, you can also look from the point of view of choosing the 5 students who will go on cars. This is C(50,5). Write it out with factorials and see that these two expressions give the same result.

Edit: I forgot to mention that, like Soroban, I chose to ignore the word "each" because it seemed to be a typo.

4. Thanks! I don't understand 2 though...

5. Originally Posted by acc
Thanks! I don't understand 2 though...
Write the left hand side in terms of factorials, and the right hand side in terms of factorials, and then solve for n. If you get stuck, post whatever work you were able to complete.