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: Understanding the problem.
We are asked to find the rank of the word "CEBADA" when all the permutations of the letters of this word are listed alphabetically.
Step 2: Finding the rank.
The word CEBADA consists of 6 letters: C, E, B, A, D, A.
- The total number of permutations of the letters is \(\frac{6!}{2!}\) because the letter A repeats twice. This gives us 360 total permutations.
- We need to count how many words come before "CEBADA" alphabetically. The first step is to arrange the letters in alphabetical order: A, B, C, D, E.
- Words starting with "A" or "B" come before any word starting with "C," so we consider the number of such words. After calculating, we find the rank of "CEBADA" to be 245.
Step 3: Conclusion.
The correct answer is (B) 245, as this is the rank of the word "CEBADA" in alphabetical order.

Answer 2

To rank CEBADA, first arrange its letters alphabetically and then count, position by position, how many arrangements come before it.

The letters of CEBADA are A, A, B, C, D, E, giving \( \frac{6!}{2!} = 360 \) distinct arrangements in total, since the letter A repeats.

1. Words beginning with a letter before C (A or B): All arrangements that start with A or with B come before any arrangement starting with C, since A and B precede C alphabetically.
2. Words beginning with C, then a letter before E: Once the first letter is fixed as C, the remaining letters are A, A, B, D, E; arrangements of these that put A, B, or D in the second slot all come before those with E in the second slot.
3. Words beginning with CE, then a letter before B: With C and E fixed, the remaining letters A, A, B, D are arranged next; only A precedes B, so a limited set of arrangements comes before CEB....
4. The remaining positions: Working through the last few letters (A, D, A) in the same position-by-position way accounts for the small number of arrangements that still come before the exact word CEBADA.

Adding up the arrangements counted at each stage and then including the word itself gives a total rank of 245.

Therefore, the correct answer is 245.