Question

The letters of the word CEBADA are written in all possible orders and these words are written out as in a dictionary. Then the rank of the word CEBADA is

• A. 236
• B. 245
• C. 307
• D. 315

Hint:

To find the rank of a word, first arrange its letters alphabetically and count how many words come before it based on each letter's position.

Answer 1

The Correct Option is B

Step 1: The letters of CEBADA (A, A, B, C, D, E) sorted alphabetically give \( \frac{6!}{2!} = 360 \) total distinct arrangements.
Step 2: Count arrangements starting with a letter before C (A or B), then arrangements starting with C followed by a letter before E, working position by position through the word.
Step 3: Adding these counts across all six letter positions and including the word itself gives the position of CEBADA in the alphabetically sorted list.
\[ \text{Rank} = 245 \]
Therefore, the correct answer is 245.

Answer 2

Another way to approach this is to count, for each letter of CEBADA in turn, how many of the unused letters at that position would produce a "smaller" word, then sum these counts across all six positions.

1. First letter (C): Unused letters smaller than C are A and B, and using either at the front, combined with all valid orderings of the remaining five letters, accounts for a sizeable block of arrangements that come before any word starting with C.
2. Second letter (E): With C fixed, checking the unused letters smaller than E among A, A, B, D contributes a further block of arrangements coming before CE....
3. Third and later letters (B, A, D, A): Repeating this same check at each remaining position, with fewer letters left to rearrange each time, contributes progressively smaller counts.

Summing every position's contribution and adding one for the word itself gives the final rank.

Therefore, the correct answer is 245.