Question

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is
A) ${{45}^{th}}$
B) ${{46}^{th}}$
C) ${{44}^{th}}$
D) ${{47}^{th}}$

Answer 1

In this, we will find the rank or the position of the word QUEEN in the English dictionary. As we have to find the position of the word in English dictionary we will find the number of the words arranged with the word QUEEN in alphabetical order. In this we will use the formula of permutation in which in n objects there, where ${{p}_{1}}$ objects are of one kind, ${{p}_{2}}$ objects are of second kind and, ${{p}_{k}}$ objects are of ${{k}^{th}}$ kind and formula is $\dfrac{n!}{{{p}_{1}}!\cdot {{p}_{2}}!\cdot \cdot \cdot {{p}_{k}}!}$

Complete step by step answer:
The alphabet of the word QUEEN in alphabetic order is E, N, Q and U.
Now we will find the number of words starting with E.
IF we start the word with E then the position remaining for the remaining alphabet are 4( in the 4 position the alphabet includes E also as it appears twice in the word QUEEN).
Hence, the number of words starting with E = 4! =24
Now we will find the number of arrange when word starting with the alphabet N.
If the word starts with N then for the remaining 4 positions it includes two E. Therefore, we will use the formula given in the hint.
Hence, the number of words starting with N = $\dfrac{4!}{2!}$=12
Now we will find the number of alphabet arrange when words starting with QE
If the words start with QE then the remaining positions are 3 with 3 remaining alphabet with E.
Hence, the number of words starting with QE = 3! = 6
Now we will find the number of alphabet arrangements when words starting with QN.
If the words start with QN then the remaining 3 positions include two E.
Hence, the number of words starting with N = $\dfrac{3!}{2!}$=3.
Now we will find the number of alphabet arranged when words starting with QU.
The word starts with QU is QUEEN in Alphabetical order
Hence the number of words with QU = 1.
The position of word QUEEN in English dictionary = 24+12+6+3+1=46
Hence the position of the word QUEEN is ${{46}^{th}}$.

So, the correct answer is “Option B”.

Note: In this problem, students should note we did not find the number of words starting with U as we have to find the position of word QUEEN. If we have to find the number of arrangements of words QUEEN then we will include the words starting with U. Try not to make any calculation errors.