Question

Consider all possible permutations of the letters of the word ENDEANOEL.

Answer 1

Nine letters : $3E^s,2N^s,4$ different $D,A,O,L$.
$ENDEA,N,O,E,L$ Five different
$\left(a\right)\to \left(p\right)$ String Method, We have five different units whose permutation is $5!$.
$\left(b\right)\to \left(s\right)E$ ENNDAOL $E$
$1st$ and last places are filled with $E^S$. We have to arrange remaining $7$ letters out of which $2N^S$ are alike.
$\therefore \frac{7!}{2!}=\frac{7.6.5!}{2}=21(5!)$
$\left(c\right)\to \left(q\right)D,L,N,N$ will not occur in last five and hence they have to be arranged in first $4$ position. $\therefore \frac{4!}{2!}$
For the last five letters out of which $3E^S$ are alike $\frac{5!}{3!}$.
Hence total $=\frac{4!}{2!}\cdot \frac{5!}{3!}=\frac{4.(3!)}{2}\cdot \frac{5!}{3!}=2(5!)$
$\left(d\right)\to \left(q\right)AEO$ to occur in odd positions. i.e., $1st,3rd,5th,7th$ and $9th$.
$\frac{5!}{3!\left(alikeE\right)}\cdot \frac{4!}{2!\left(alikeN\right)}=5!\frac{4}{2}=2(5!)$