Combinatorial Problem of Letters

[Seydlitz]

Homework Statement

From this combination of letters AAAXYZNO
Find how many ways to pick 3 letters if the order does not matter.

The Attempt at a Solution

I tried to elaborate it like this:

We have ___ 3 empty spaces.

A__ (Two empty space for other different letters) -> 5C2 = 10
AA_ (One empty space) -> 5C1 = 5
AAA (All AAA) -> 1 way only.
___ (No A) -> 5C3 = 10

The consideration is __A and A__ will be just the same because the order does not matter.

Hence, total ways = 26.

Or another way is simply 6C3 because we eliminate all of the As giving 20 ways only.

I also want to ask if you guys know the insight that can be shared in solving this kind of problem since I feel the concept is just floating in my mind without concrete standing.

 

[haruspex]

26 is right, and that is probably the best method.
I did not understand the logic behind 6C3. You could use that to count all the ways with at most one A, I suppose,but you still need to add in the two and three A cases.

 

[lurflurf]

The three way I see are
5C0+5C1+5C2+5C3=1+5+10+10=26
6C1+6C3=6+20=26
8C3-6C2-6C2=8C3-6P2=56-30=26
whichever you like best