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## Homework Statement

I forget the exact expression of the questions. But the related details are exhaustive here.

it's about permutation and combination. By the way, I am not student, i am looking for explanation and understanding, not answers..

1. There are 6 people, namely A, B, C, D, E, F in a row, how many ways of arrangement are there such that person A,B,C must be separated

## Homework Equations

No equation given

## The Attempt at a Solution

I especially paid attention to the word "must be separate", then I first count the number of ways that A,B and C "must be grouped" together. which result in 3! (arrangement within A,B,C) and then multiply 4! (taken ABC as one object).

Then for 6 people randomly arranged in order there are 6!. It's easy.

so the solution i think should be 6! - 3! x 4!

however, that was wrong. the solution is 3! x 4!

I feel so hard understanding why.

I don't know what they mean by "must be separate" here, how about, for example, A E F B C D? when B and C are combined while A is separate from them. Did I forgot to minus this possibility?

The answer is actually 144 but I get no idea about it.

I am not really clever, please explain in a simple way. Thanks people. Thanks people