Calculate the numbers of ways to arrange 6 letters out of 7 letters of

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In summary, there are 210 ways to arrange 6 letters out of the 7 letters in the word 'SETTEES'. This is calculated by finding the number of ways to arrange 3 E's, 2 S's, and 2 T's and then dividing by 24 to account for equivalent starting configurations. This can be done using the multinomial theorem or by going through the mental process of choosing an arrangement and counting the choices at each stage.
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xender_c
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Calculate the numbers of ways to arrange 6 letters out of 7 letters of the word `SETTEES'.
Answer:210
Could anyone explain this to me ?thx
 
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try making different cases, for example 3 alike of 1 kind 2 alike of another kind and 1 of third kind.
or if you have read multinomial theorem try using that.it is simpler.
 
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I find it most helpful to go through the mental process of actually choosing an arrangement and counting the choice one makes at each stage. You multiply the numbers for each stage to get the total multiplicity of choices. You then consider if you created any equivalent arrangements and divide those out. Example: Arrangements of two of the letters "ADD".

First I would distinguish the two D's { A, D1, D2}.
There are three letters and I have 3 choices for the first in my arrangement. There are two remaining and I have 2 choices for the 2nd. That's 3 x 2 = 6 choices.

Now since we can't really distinguish D1 vs D2 we divide out the ways we may have multiply chosen the same arrangement.

Of those 6 arrangements starting with any of the 2 permutations of D1 and D2 will give me an identical sequence of choices. So I divide by 2!=2 6/2 = 3.
AD DA and DD.

In your case there are 3!=6 ways to permute the three E's times 2!=2 ways to permute the S's and 2!=2 ways to permute the T's. That's 6x2x2=24 equivalent starting configurations. Figure the way to arrange 6 of 7 distinct letters and then divide by 24.
 

What is the formula for calculating the number of ways to arrange 6 letters out of 7 letters?

The formula for calculating the number of ways to arrange 6 letters out of 7 letters is 7P6 = 7!/1! = 7.

How many combinations are there when arranging 6 letters out of 7 letters?

There are 7 combinations when arranging 6 letters out of 7 letters. This can be calculated using the formula 7C6 = 7!/6!1! = 7.

What is the difference between permutations and combinations?

Permutations refer to the number of ways to arrange a set of objects in a specific order, while combinations refer to the number of ways to select a subset of objects from a larger set, regardless of their order.

Can the order of the letters be repeated in the arrangements?

Yes, the order of the letters can be repeated in the arrangements. This is known as repetition or with replacement, where each letter can be used more than once in a single arrangement.

How do you calculate the number of ways to arrange 6 letters out of 7 letters if repetition is not allowed?

If repetition is not allowed, the number of ways to arrange 6 letters out of 7 letters can be calculated using the formula nPr = n!/(n-r)!, where n is the total number of letters (7) and r is the number of letters to be arranged (6). This results in 7P6 = 7!/(7-6)! = 7!/1! = 7.

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