The letters "CFOSU" are arranged in dictionary order. What is the rank

It would be highly appreciated if someone could elaborate a little bit on this. Thanks

Solution

Given
In this question, we are given that

• All the letters of “CFOSU” are arranged in dictionary order

To find
We need to determine

• The rank of the word “FOCUS”

Approach and Working out
If all the letters are arranged in alphabetical order, then we will have words starting with C first, then F, then O, then S, and finally U.

If C is the first word, then the remaining words can arrange themselves in 4! = 24 ways

Next, we will have words starting with F.

• Words starting with FC = 3! = 6
• Words starting with FOC: FOCSU and FOCUS

Hence, the rank of the word FOCUS = 24 + 6 + 1 + 1 = 32

Thus, option D is the correct answer.

Correct Answer: Option D

Can anyone recommend more faster or alternative approach here please?

Words starting with letter C = 4! = 24
With F as first letter, words starting with C as second letter = 3! = 6
Words with F as first and C as second letter, O as third letter, S as fourth and U as fifth letter = 1
Ranking of word FOCUS = 24 + 6 + 1 + 1 = 32

IMO D

The first word in this order will be CFSOU

With C as starting letters, we have 4! ways = 24 arrangement.
With F as starting letters, we have FSxxx before FOxxx, so 3! = 6 more numbers before FOCUS.
With FO as starting letters, clearly C is the next one to go. So we have FOCxx.

Since U is behind S in the alphabet, we have FOCSU and then FOCUS. The rank of FOCUS is 24+6+1+1 = 32.
IMO D.

I hate this type of question so much. Singing the alphabet song to solve this kind of problems takes a huge toll on my manhood.

Once the nuance of the game is understood it's pretty simple
CFOSU = 4! ways
FCOSU = 3! ways
FOCSU = 1 and FOCUS ; 1
total 4!+3!+1+1 ; 32
IMO D

Let's assign ranks to each letter based on their alphabetical order:

C F O S U

1 2 3 4 5

Now, for words starting with C (the first ranking letter) the remaining words can be arranged in 4! ways = 24 ways (since we know C's position is fixed).

Now, for the next ranking letter, F, the following letter should be C, because we are looking at alphabetical order. Thus, given that F and C are fixed, the remaining 3 letter can be arranged in 3! ways = 6 ways.

Note, that we are making our way down the list by taking factorials for each arrangement.

Now we have to look at words starting with FO. When F O are in a fixed position, the next letter to consider should be C, since we are following the alphabetical order. When F O C are in a fixed position, the remaining two letters can arrange themselves in 2! ways = 2 ways.

Note that FOCUS will be the last word if we follow the alphabetical order, since after F O C, S should come before U creating the word FOCSU. After that, only FOCUS remains.

Thus to reach FOCUS we had to go through 24 + 6 + 2 = 32 words. Thus, FOCUS is on the 32nd position.