Combinatorial Problem of Letters

  • Thread starter Seydlitz
  • Start date
In summary, the total number of ways to pick 3 letters from the combination AAAXYZNO where the order does not matter is 26. This can be calculated by using the combinations formula and considering all possible combinations with 0, 1, 2, or 3 A's. Another method is using 6C1+6C3, which eliminates all the A's and counts the remaining combinations. It is also possible to use 8C3-6C2-6C2, which takes into account the fact that the order does not matter and also eliminates the A's from the calculation.
  • #1
Seydlitz
263
4

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.
 
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  • #2
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.
 
  • #3
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
 

What is the Combinatorial Problem of Letters?

The Combinatorial Problem of Letters is a mathematical problem that involves finding all possible combinations of letters from a given set of letters. It is often used in cryptography, genetics, and computer science.

How is the Combinatorial Problem of Letters solved?

The Combinatorial Problem of Letters can be solved using various methods, such as brute force, recursion, and dynamic programming. The most efficient method depends on the specific problem and constraints.

What are the applications of the Combinatorial Problem of Letters?

The Combinatorial Problem of Letters has various applications, including creating strong passwords, generating new words for languages, and identifying patterns in genetic sequences.

What are some common mistakes when solving the Combinatorial Problem of Letters?

One common mistake is not considering all possible combinations, leading to an incorrect solution. Another mistake is not properly defining the problem or its constraints, resulting in an incomplete or incorrect solution.

How can the Combinatorial Problem of Letters be made more efficient?

The Combinatorial Problem of Letters can be made more efficient by using optimization techniques, such as pruning unnecessary branches in the search tree, or by using parallel processing to explore multiple combinations simultaneously.

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