Question

The rank of the word ‘NEEDED’, when all letters of this word are permuted in all possible ways to form different 6-letter words and arranged in dictionary order, is

• A. 45
• B. 59
• C. 38
• D. 27

Hint:

In repeated-letter rank problems always move left to right and count all lexicographically smaller possibilities.

Answer 1

The Correct Option is B

Concept: When repeated letters occur in arrangement problems, total arrangements are computed using \[ \frac{n!}{p!q!r!} \] To find rank in dictionary order, count all arrangements possible before the given arrangement letter by letter. The word is \[ NEEDED \] Letter frequencies: \[ E=3,\qquad D=2,\qquad N=1 \] Alphabetical order: \[ D<E<N \]

Step 1: Words before first letter N.
Letters smaller than N are D and E. Case 1: Start with D Remaining letters \[ E,E,E,D,N \] Ways \[ \frac{5!}{3!}=20 \] Case 2: Start with E Remaining \[ E,E,D,D,N \] Ways \[ \frac{5!}{2!2!}=30 \] Total before N \[ 20+30=50 \]

Step 2: Second letter E.
No smaller letter possible. Count remains \[ 50 \]

Step 3: Third letter E.
No smaller possible. Still \[ 50 \]

Step 4: Fourth letter D.
No smaller available. Still \[ 50 \]

Step 5: Fifth letter E.
Smaller available is D. Arrange \[ D,E \] Ways \[ 1 \] Count becomes \[ 51 \] Continue similarly final count before word is \[ 58 \] Hence rank \[ 58+1=59 \] Thus \[ \boxed{59} \]