Arranging letters with two letters not next to each other

just wanna know what’s the answer to this one.

How many ways can the word “ARRANGED” be arranged so that N and D aren’t next to each other?

Thanks. 🙂

Solutions Collecting From Web of "Arranging letters with two letters not next to each other"

The total number of words is $\binom82\cdot\binom62\cdot\binom41\cdot\binom31\cdot\binom21\cdot\binom11=10080$.

The number of words with ND is $\binom72\cdot\binom52\cdot\binom31\cdot\binom21\cdot\binom11=1260$.

The number of words with DN is $\binom72\cdot\binom52\cdot\binom31\cdot\binom21\cdot\binom11=1260$.

Hence the number of words with no ND and no DN is $10080-1260-1260=7560$.

Notice, in the given word $ARRANGED$ there are total $8$ letters out of which $A$ & $R$ are repetitive letters hence, we have

Total number of linear arrangements of $ARRANGED$ $$=\frac{8!}{2!2!}$$
Number of linear arrangements when $N$ & $D$ are kept together $$=\frac{7!}{2!2!}\cdot 2!=\frac{7!}{2!}$$
Hence, the number of linear arrangements when $N$ & $D$ are not kept together
$$=\text{(total number of linear arrangements)}-\text{(number of linear arrangements when N & D are kept together)}$$
$$=\frac{8!}{2!2!}-\frac{7!}{2!}=\frac{3\cdot 7!}{2!}=\color{red}{7560}$$