Question

All the rearrangements for the letters of the word ${}^{\prime }DEMAN{D}^{\prime }$ are written without including any word that has two $D^{\prime }$s appearing together. If all these are arranged in dictionary order, then the rank of the word $"DEMAND"$ will be

• A. $36$
• B. $74$
• C. $42$
• D. $86$

Answer 1

The correct option is B $74$

Alphabets present in the word $DEMAND$ are $D,E,M,A,N,D$
Dictionary Order of these letters is $A,D,D,E,M,N$
Words starting with $A:\frac{5!}{2!}=60$
Words starting with $A$ where two $D^{\prime }s$ are together $=4!=24$
$\therefore$Words starting with ${}^{\prime }{A}^{\prime },$ without two $D^{\prime }s$ adjacent to each other $=60-24=36$

Next we have words starting with ${}^{\prime }{D}^{\prime }$.

Within this, we have words starting with $DA\left(DA\_\_\_\_\right)$: $4!$ words $=24$ words
Then words starting with $DE$
Within this, words starting with $DEA\left(DEA\_\_\_\right)\Rightarrow 3!=6$ words

Then starting with $DED\left(DED\_\_\_\right)\Rightarrow 3!=6$ words
Then starting with $DEM$
$\Rightarrow$ First word is $DEMADN$
$\Rightarrow$ Second word is $DEMAND$

$\therefore$Rank of the word $DEMAND=36+24+6+6+2=74$