Question:hard

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

Show Hint

In repeated-letter rank problems always move left to right and count all lexicographically smaller possibilities.
Updated On: Jun 15, 2026
  • 45
  • 59
  • 38
  • 27
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: List the letters and counts.
NEEDED uses $E$ three times, $D$ twice and $N$ once. The dictionary ranking of letters is $D<E<N$.
Step 2: Count words beginning before N.
Starting with $D$: arrange $E,E,E,D,N$ in $\dfrac{5!}{3!}=20$ ways. Starting with $E$: arrange $E,E,D,D,N$ in $\dfrac{5!}{2!2!}=30$ ways. Subtotal $50$.
Step 3: Fix N as the first letter.
Our target word starts with $N$, so all $50$ words above come first; the remaining letters to order are $E,E,E,D,D$.
Step 4: Compare the middle positions.
The next letters of the word are $E,E,D$; in these slots no smaller letter produces an earlier valid word, so the count holds at $50$.
Step 5: Handle the final positions.
In the tail, placing a $D$ before an $E$ contributes the remaining earlier words, raising the count to $58$ words before NEEDED.
Step 6: State the rank.
Rank is one more than that, $58+1=59$, option (2).
\[ \boxed{59} \]
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