### Instruction: My grandpa has 10 pieces of art, including 3 prints by Escher. If he hangs the pieces of art in a row in a random order, what is the probability that all three pieces by Escher will be placed consecutively?### Response: This is a probability problem that involves counting outcomes.
I need to find the total number of ways to arrange 10 pieces of art in a row, and the number of ways to arrange them such that the 3 pieces by Escher are together.
To arrange 10 pieces of art in a row, I can use the formula for permutations, which is n!, where n is the number of objects.
In this case, n = 10, so the total number of arrangements is 10!.
To arrange them such that the 3 pieces by Escher are together, I can think of the 3 pieces as one unit, and then arrange the 8 units in a row.
The number of ways to do this is 8!, since there are 8 units.
However, this is not the final answer, because within the unit of 3 pieces by Escher, there are 3! ways to arrange them.
So I need to multiply 8! by 3! to get the number of ways to arrange them such that the 3 pieces by Escher are together.
Therefore, the number of ways to arrange them such that the 3 pieces by Escher are together is 8! * 3!.
The probability of this happening is the ratio of the number of favorable outcomes to the number of possible outcomes.
So the probability is (8! * 3!) / 10!, which I can simplify by canceling out some factors.
I get (8! * 3!) / 10! = (3 * 2 * 1) / (10 * 9) = 6 / 90 = 1 / 15.