More perms and combs

[andy62]

For some obscure reason I need to find out how many different arrangements of 4 letters I can choose from the the letters of SLEEPY SENATE.

Once again I find myself in disgreement with the book.

As I see it there are 5 cases.

4xE=1 way

3xE +1=28 ways

2xE+2xS=6 ways

All different=1680 ways

The last arrangement would be 2 alike letters plus 2 others.

I have broken this up into

2xS+1xE+1
2xS+2 from 6 (no E)
2xE+1XS+1
2xE+2 from 6 (no S)

which gives me 201 arrangements.

The book I'm using has a total of 504 arrangements for this part.

Who is right?

[mohill]

Hi andy62! I agree with the book.
4E = 1
3E + 1 = 28
2E + 2S = 6
2E + 2different from each other = 6x7x6 = 252
2S + 2different from each other = 6x7x6 =252
4different from each other = 8x7x6x5 =1680
Total = 2219

You may have omitted the second to last possibility

[Sonyme]

There is a formula for this type of calculation which would be
12*11*10*9/(1*2*3*4) =495

Check any bookmakers diaries giving no of multiple bets fr any number of selections

This would equate to how many fourfold accumulators from 12 selections

[andy62]

Thanks Mohill

I can see your logic (by the way I really do not like this topic) but I am struggling to see how you can treat the multiple S's or E's as single entities.

Most discombobulated!

Andy

[mohill]

4E there is obviously only 1 way they could be arranged

3E + 1 The 1 could be placed in any one of the 4 positions and in each case the 3Es take the three places remaining.
But the 1 could have been chosen from any of the 7 possible letters. Hence 4 x 7 = 28 arrangements.

2E + 2different from each other. 2Es can be placed in 6 different ways (see below). There are then 7 choices for th 1st 'different' and once it has been placed there are 6 choices for the 2nd 'different' one. So 6 x 7 x 6 =252

2E + 2S It is easier to see the permutations rather than use maths logic.
EESS ESES ESSE SESE SSEE SEES So 6 cases

2S + 1different
The logic is exactly the same as for 2E + 2different above.
Hence 6 x 7 x 6 = 252

All different from each other.
8 choices for the 1st position, then 7 , then 6 , then 5.
8 x 7 x 6 x 5 = 1680

1 + 28 + 252 + 6 + 252 + 1680 = 2219

I hope this helps.

[andy62]

Thanks again mohill. Now it is clear.