Question

If the letters of the word APPLE are arranged as in a dictionary, the rank of the word APPLE is:
(a) 23
(b) 27
(c) 19
(d) None of these

Answer 1

Hint: Find out the number of words that can be formed with the starting letters as AE, AL, APE, APL, APPE. Now, the next word will be APPLE. So, after finding the number of words form with letters AE, AL, APE, APL and APPE add 1 to get the rank.

Complete step by step answer:
Here, we have been provided with the word APPLE and we have to find its rank if it is arranged as in the dictionary.
Now, APPLE will come after all the words formed by the starting letters AE, AL, APE, APL and APPE. So, let us find the total number of words that can be formed using these as starting letters.
(i) Considering AE as the first two letters. Now, we have three letters remaining which are P, P and L. P is occurring twice.
\[\therefore \] Number of words starting with AE = \[\dfrac{3!}{2!}=3\]
(ii) Considering AL as the first two letters. Now, we have three letters remaining which are P, P and E. P is again occurring twice.
 \[\therefore \] Number of words starting with AL = \[\dfrac{3!}{2!}=3\]
(iii) Considering APE as the first three letters. Now, we have two letters remaining which are P and L.
\[\therefore \] Number of words starting with APE = 2! = 2
(iv) Considering APL as the first three letters. Now, we have two letters remaining which are P and E.
\[\therefore \] Number of words starting with APL = 2! = 2
(v) Considering APPE as the first four letters. Now, we have one letter left which is L.
\[\therefore \] Number of words starting with APPE = 1! = 1
Now, the next word will be APPLE. So, the number of words before APPLE = 3 + 3 + 2 + 2 + 1 = 11.
So, the rank of APPLE as arranged in the dictionary will be the sum of the number of words before APPLE and 1.
\[\Rightarrow \] Rank of APPLE = 11 + 1 = 12

So, the correct answer is “Option D”.

Note: It is important to note that if a letter is repeating ‘n’ times then we have to divide the arrangement with n!, just like ‘P’ was repeating 2 times in case (i) and (ii), so we divided the total arrangement of letters with 2!. Note that whenever rank of a word is asked you must add 1 to the number of words that are formed before that particular word.