Suppose you have six different books on a shelf with labels $A, B, C, D, E,$ and $F$.
In how many different ways can you arrange the books on the shelf if books $A, B,$ and $C$ are grouped together?
(e.g. $EBACFD$ is an acceptable arrangement, but $EBAFDC$ is not an acceptable arrangement)
While trying to solve it I get $120$, by $6\times5\times2\times1\times2\times1$. Although the correct answer is $144$.
Can anyone correct me on where I am going wrong?
Best Answer
Let $A,B,C$ be considered a single element$=M$ (say)
So we have four elements-$M,D,E,F$
These $4$ elements can be arranged in $4!$ ways.
And the $3$ elements in $M$ can be arranged in $3!$ ways.
So the required number of ways = $4! \times 3! = 24 \times 6 = 144$