Combination usage in the well-known word problem

[M. next]

How many different letter arrangement can be made from the 11 letters of MISSISSIPPI?

(But using COMBINATION not the different permutation method)

I saw an answer and it says:
(combination of 1 out of 11)*(combination of 4 out of 10)*(combination of 4 out of 6) *(combination of 2 out of 2)

What does combination exactly mean, and in what way was it used here?

I mean if we don't want same words to appear, what will this have to do with taking say 4 Ss out of the left 10?

Thanks in advance.

 

[tiny-tim]

Hi M. next!

What does combination exactly mean, and in what way was it used here?

I mean if we don't want same words to appear, what will this have to do with taking say 4 Ss out of the left 10?

"combination of k out of n", also written "n choose k", or ⁿC_k, or the vertical matrix (n k), and equal to n!/k!(n-k)!,

is the number of ways of selecting k things out of n where the order does not matter

see http://www.mathsisfun.com/combinatorics/combinations-permutations.html for details ​

 

[D H]

(But using COMBINATION not the different permutation method)

I saw an answer and it says:
(combination of 1 out of 11)*(combination of 4 out of 10)*(combination of 4 out of 6) *(combination of 2 out of 2)Thanks in advance.

That's the answer for the number of permutations, not the number of combinations.

The difference between the two: "issi" and "siis" are two different permutations (arrangements) of the letters i,i,s, and s. They are not different combinations, however; order doesn't matter in combinations.

Suppose we put labels on the duplicated letters, making mississippi become mi₁s₁s₂i₂p₁p₂i₃s₃s₄i₄. There are 11 factorial (11!) permutations of these labeled letters because each (labeled) letter is distinct. For example, i₁s₁s₂i₂ and i₁s₁s₃i₂ are distinct permutations. Take those labels away and these two permutations become issi and issi: they are now indistinguishable.

The problem is to find the number of distinguishable permutations of the given (unlabeled) letters.

 

[M. next]

oh thanks, and sorry for the late reply

 

[CellCoree]

Combination is a mathematical term that refers to the number of ways a group of items can be selected from a larger set without regard to order. In this context, it is being used to determine the number of different letter arrangements that can be made from the 11 letters in the word "MISSISSIPPI" without considering the order in which the letters appear.

In the given solution, the combination method is being used to determine the number of ways to select a certain number of letters from the 11 letters in the word. For example, "combination of 1 out of 11" means selecting 1 letter from the 11 available letters. This is done for each letter in the word, resulting in the following combinations: 1 out of 11, 4 out of 10, 4 out of 6, and 2 out of 2.

The reason for taking 4 Ss out of the left 10 is to ensure that only unique combinations are counted. If we did not do this, we would end up with duplicate combinations such as "SSSS" or "SSPP". By limiting the number of Ss to 4 out of the left 10, we are essentially saying that we only want to consider combinations where each letter appears no more than 4 times.

Overall, the combination method is a useful tool in solving word problems like this one, as it allows us to determine the number of ways to select items without regard to their order.