If the letters a, A, b, B, c, and C are arranged at random in a row

hey Bunuel
what's wrong with my approach?
A,a,b,B,c,C
desired mode:abc
total :6!
p=4!/6!

The problem with both solutions above is that favorable outcomes are much more, namely 120.

{***}{a}{b}{c} - \(4*3!=24\) (capital letters are together);
{**}{a}{*}{b}{c} - \(C^2_3*2*4*3=72\) (2 capital letters are together);
{*}{a}{*}{b}{*}{c} - \(C^3_4*3!=24\) (capital letters are separated);

\(24+72+24=120\) --> \(P=\frac{120}{6!}=\frac{1}{6}\).

But this is a long way of solving. Consider one particular arrangement: A*B*C*, lower case letters for *. We can arrange lower case letters instead of * in 3!=6 ways but only one will be in alphabetical order AaBbCc, so 1 out of 6. For other such cases also only one out of 6 will be in alphabetical order (ABCabc, ....), so P=1/6.

Basically we can ignore capital letters for this problem and say: 3 letters a, b, and c can be arranged in 3!=6 different ways and only one of them will be in increasing alphabetical order, namely: a-b-c, so P=1/6.

Hi Bunuel,

Even i also think that answer should be 4!/6!

Think of these arrangements as the following :

___[]___[]___[]____


Where [] act as placeholders for the smaller case alphabets. And for each arrangement of small case letters, the ___ are used as placeholders for one of more upper case letters.

For Eg. The arrangement of small case letters __b__a__c___ can then produce several final arrangements such as ABCbac or bACaBc etc etc

Now all I am going to say is that for each arrangement of type ___b___a___c___ there are an equal number of final arrangements possible. As you just change the small case letters used in your placeholders keeping the upper case ones constant.

How many types of arrangement exist ? The number of ways to place small case letters in place holders which is 3!
How many of these have a,b,c in order ? Just 1 : __a__b__c__

So probability = 1/6

I am totally confused

Bunuel can you pls simplyfy the answer for me

Thanks in advance..

Can you please tell what exactly didn't you understand?

Simplifying: you can ignore capital letters for this problem. So, 3 letters a, b, and c can be arranged in 3!=6 different ways and only one of them will be in increasing alphabetical order, namely: a-b-c, so P=1/6.

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