Question

Alice has 10 balls (all different). First, she splits them into two piles; then she picks one of the piles with at least two elements, and splits it into two; she repeats this until each pile has only one element. (a) How many steps does this take? (b) Show that the number of different ways in which she can carry out this procedure is (10 choose 9) * (9 choose 8) * (8 choose 7) * ... * (3 choose 2) * (2 choose 2)

Answer 1

It takes 9 steps for Alice to split the 10 balls into individual piles. The number of different ways she can perform this task is the product of combinations at each step: (10 choose 9) * (9 choose 8) * ... * (2 choose 2).

Step-by-step explanation:

Alice is conducting a procedure where she splits a group of 10 different balls into individual piles by repeatedly dividing them. Let's break down the process step by step:

1. Alice starts with 10 balls and creates two piles. The number of ways to do this is (10 choose 9).
2. She then takes the pile with more than one ball (there are 9 in this case) and splits it into two, which can be done in (9 choose 8) ways.
3. This process is repeated, always splitting a pile with at least two balls into two piles until each pile only has one ball. So, the next split gives us (8 choose 7) ways, and so on until there is only one way to split the final two balls, which is (2 choose 2).

To answer part (a), this process takes exactly 9 steps because each step reduces the number of multi-ball piles by one until only single-ball piles remain.

For part (b), the number of different ways in which she can carry out this procedure is the product of the combinations at each step, which is (10 choose 9) * (9 choose 8) * (8 choose 7) * ... * (3 choose 2) * (2 choose 2).