Probablity Questions?

Question

1. In how many ways can four identical orange chips and two identical blue chips be arranged in a circle?
2. There are 5 teachers. Teacher A, B, C, D, E. How many possible seating arrangement are there if teacher A and B cannot sit next to each other?

Answer 1

1. In how many ways can four identical orange chips and two identical blue chips be arranged in a circle?

This is small enough that you can just count them.

i. The two blue chips are next to each other.

ii. There is one space between them the short way and 3 spaces between them the long way.

iii. There are two spaces between them in both directions.

The answer is 3.
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2. There are 5 teachers. Teacher A, B, C, D, E. How many possible seating arrangement are there if teacher A and B cannot sit next to each other?

You didn't say how the teachers are arranged. Are they in a circle or a straight line? I will assume they are in a straight line since you didn't say.

The total number of arrangements is 5! = 120. We need to subtract from that number the arrangements in which A and B are next to each other.

Answer = 5! - (2*4)*3! = 120 - 8*6 = 72

Answer 2

1. There are only 3 possibilities. The two blue chips can either be next to each other, or have one orange chip between them, or be opposite each other (with two orange chips between them).

2.If they are sitting in a row, I think there are 72 possible arrangements. (There are 6 possibilities with each teacher on the left of the other three (not counting B), so that's 24. Then each of those has 3 places where B can sit and not be next to A. 3x24=72