I dislike combinatorial questions because I am not a native English speaker, and this entire field uses non-rigorous language, such as "placing", "choosing", "taking away" and uses playful examples like poker game, which uses words like "diamond" or "spade" that are foreign to me.
The biggest challenge for me right now is to trying to understand why we divide instead of subtract in doing combinatoric questions.
Example: A farmer is planting 5 red flowers, 3 yellow flowers, and 2 white flowers in a row. How many different ways can the farmer plant these flowers?
Ok, so I answer this question like this:
Imagine we have a list of boxes that we can place these flowers in:
[][][][][][][][][][]
The first box can take on 5+3+2 = 10 flowers, next 9....so in total we have 10*9*8....*1 = 10! possible ways, if all flowers are labelled.
Since these flowers are not labelled, now I need to "take away" those "repeated" flowers. ("Taking away" - At least this is what teachers or textbook usually say)
Ok, so what does "take away" mean? Thought experiment:If I had 5 apples, I take away 3, I have 2 left. Aha! So 5 - 3 = 2. Taking away means subtract!
Since each of these 5 red flowers can be arranged in 5*4*3*2*1 = 5! ways, yellow flowers in 3! ways, and white flowers in 2! ways.
Therefore, the total number of ways the farmer can arrange these flowers is 10!-5!3!2!.
Why am I wrong in this approach? Why is the correct approach is to divide 10! by 5!3!2!?
I always think of it this way:
n! = number of unique permutations * number of times each unique permutation is copied
where n is the number of objects you are permuting. So if we divide n! by the number of times each permutation is copied, we get the right number. In your case, n=10 and each unique arrangement of flowers is copied 5!*3!*2! times