SOLUTION: a pile of 11 coins is to be separated in piles of 2,3 and 6 coins. How many ways are there to do this? explain.

 
1. Suppose we pick the pile of 2 first, the pile of 3 second, and the pile of 6 third. 

Choose the pile of 2 from the 11 coins in 11C2 = 55 ways
Choose the pile of 3 from the remaining 9 coins in 9C3 = 84 ways
Choose the pile of 6 from the remaining 6 coins in 6C6 = 1 way.
That's (55)(84)(1) = 4620 ways.

The answer is 4620.  

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But just for fun, let's suppose we formed the piles in a different order.

2. Suppose we pick the pile of 2 first, the pile of 6 second, and the pile of 3 third. 

Choose the pile of 2 from the 11 coins in 11C2 = 55 ways
Choose the pile of 6 from the remaining 9 coins in 9C6 = 84 ways
Choose the pile of 3 from the remaining 3 coins in 3C3 = 1 way.
That's (55)(84)(1) = 4620 ways. 

Or this order:

3. Suppose we pick the pile of 3 first, the pile of 2 second, and the pile of 6 third. 

Choose the pile of 3 from the 11 coins in 11C3 = 165 ways
Choose the pile of 2 from the remaining 8 coins in 8C2 = 28 ways
Choose the pile of 6 from the remaining 6 coins in 6C6 = 1 way.
That's (165)(28)(1) = 4620 ways. 

Or this order:

4. Suppose we pick the pile of 3 first, the pile of 6 second, and the pile of 2 third. 

Choose the pile of 3 from the 11 coins in 11C3 = 165 ways
Choose the pile of 6 from the remaining 8 coins in 8C6 = 28 ways
Choose the pile of 2 from the remaining 6 coins in 2C2 = 1 way.
That's (165)(28)(1) = 4620 ways. 

Or this order:

5. Suppose we pick the pile of 6 first, the pile of 2 second, and the pile of 3 third. 

Choose the pile of 6 from the 11 coins in 11C6 = 462 ways
Choose the pile of 2 from the remaining 5 coins in 5C2 = 10 ways
Choose the pile of 3 from the remaining 3 coins in 3C3 = 1 way.
That's (462)(10)(1) = 4620 ways.

Or this order:

6. Suppose we pick the pile of 6 first, the pile of 3 second, and the pile of 2 third. 

Choose the pile of 6 from the 11 coins in 11C6 = 462 ways
Choose the pile of 3 from the remaining 5 coins in 5C3 = 10 ways
Choose the pile of 2 from the remaining 2 coins in 2C2 = 1 way.
That's (462)(10)(1) = 4620 ways.  

So, as you see, it doesn't matter in which order we pick the 3 piles, the
answer is always 4620.  In some cases, we have different numbers to
multiply, but the answer always comes out to 4620 ways.

EdwinM