Arrangement of letters in a word

[rajeshmarndi]

Homework Statement

In how many ways 4-lettered words that can be formed using the letters of the word "BOOKLET".

The book solution solve it this way.
The word contains 7 letters out of which there are 2 O's. So there are 6 letters.
∴ The number of 4 lettered words
= ⁷P₄ - ⁶P₄ = 840 - 360 = 480

Homework Equations

The Attempt at a Solution

What we need is to deduct the duplicate counting of 2 O's from the total permutation.
So how does ⁶P₄ solve this purpose.

Instead of ⁶P₂,
isn't it should have been
( ⁴P₂ * ⁵P₂ ) / 2
i.e number of ways these 2 O's can be placed in 4 place * number of ways B,K,L,E,T can be placed in the remaining 2 places divide by 2.

= (12 * 20)/2 =120

And the answer would be 840 - 120 = 720 and not 480.

 

[Orodruin]

It is not only the words with two Os that are double counted in ⁷P₄.

 

[rajeshmarndi]

Instead of ⁶P₂
isn't it should have been

Correction it is ⁶P₄

It is not only the words with two Os that are double counted in ⁷P₄.

But I do not see any other letters being doubled.
My question, is the book solution, correct, any way?

 

[haruspex]

But I do not see any other letters being doubled.

Imagine the two Os in different colours. In all the words corresponding to ⁷P₄, which words only differ in colour pattern?
(I agree with the book answer, though I've yet to understand how they obtain it by subtracting ⁶P₄.)

 

[rajeshmarndi]

Imagine the two Os in different colours.

I understood that and that's why I mentioned them as O1 & O2.

In all the words corresponding to ⁷P₄, which words only differ in colour pattern?

I am not sure if I've understood you correctly, if we imagine the two Os in different colours(and each unique letter to be a different colour), then ⁷P₄ will result all to be in different colour.[/QUOTE]

(I agree with the book answer, though I've yet to understand how they obtain it by subtracting ⁶P₄.)

Where am I wrong in my below solution.

Instead of ⁶P₂,
isn't it should have been
( ⁴P₂ * ⁵P₂ ) / 2
i.e number of ways these 2 O's can be placed in 4 place * number of ways B,K,L,E,T can be placed in the remaining 2 places divide by 2.

= (12 * 20)/2 =120

And the answer would be 840 - 120 = 720 and not 480.

 

[haruspex]

Where am I wrong in my below solution.

Consider BLO₁K, BLO₂K.

 

[Orodruin]

Imagine the two Os in different colours. In all the words corresponding to ⁷P₄, which words only differ in colour pattern?
(I agree with the book answer, though I've yet to understand how they obtain it by subtracting ⁶P₄.)

I called them O and O' in my mid, the principle is the same. I too have not understood how they arrive at it being ⁶P₄ other than numerical coincidence. I solved it by first considering all words as double counted and then adding those that this removes by error (i.e., half of the 4 letter words made from BKLET).

 

[rajeshmarndi]

Consider BLO₁K, BLO₂K.

Yes, thanks, it just didn't hit my mind.

 

[rajeshmarndi]

But, I am still not yet clear.

As ⁷P₄ - ⁶P₄ = number of words containing 2 Os (including O1 & O2)

, since ⁶P₄ = total permutation without 2 Os in the word.

 

[haruspex]

But, I am still not yet clear.

As ⁷P₄ - ⁶P₄ = number of words containing 2 Os (including O1 & O2)

, since ⁶P₄ = total permutation without 2 Os in the word.

Yes, but it's coincidence. 360 have no Os, 240 have one O, 180 have two. It just happens that 360=(240+180)/2.
I can make an argument for subtracting 4⁶P₃. Consider the number that have a specific O (whether or not they have both).

 

[Orodruin]

360 have no Os, 240 have one O, 180 have two.

5! = 120 has no Os, 4*5*4*3 = 240 has one O, (4!/(2!)^2)*5*4 = 120 has two Os. The sum of all of these is 480, alternatively you subtract 240+120 = 360 from 840 and get the same result.

360=(240+180)/2

Average of 240 and 180 is 360?

 

[haruspex]

5! = 120 has no Os, 4*5*4*3 = 240 has one O, (4!/(2!)^2)*5*4 = 120 has two Os. The sum of all of these is 480, alternatively you subtract 240+120 = 360 from 840 and get the same result.Average of 240 and 180 is 360?

I shouldn't post while watching Le Tour.