Question

The letters of the word ‘GUITAR’ are arranged in all possible ways and these words are written out as in dictionary, then the word ‘GUITAR’ appears at the serial number
A. 225
B. 229
C. 227
D. 494

Answer 1

Hint:As words in dictionary are arranged in alphabetical order, we need to check all the words beginning with letters in alphabetical order. In such problems basic probability concepts will be used like factorial, permutation and combination. Factorial of a number greater than or equal to zero will be the product of all numbers less than or equal to that number ($n! = n \times (n - 1)!$). Permutation is being used in situations like arranging n objects in r positions. Combination is also used for the same but the order of arrangement does matter.

Complete step-by-step answer:
First of all let us separate the word GUITAR into letters and do the dictionary order arrangement. Then the letters in that particular order will be A, G, I, R, T, U.
As we need to find the serial number of the word GUITAR, there comes words starting with A at first.
Thus we need to count the number of words starting with A in GUITAR.
Words starting with A= Words fixing letter ‘A’ in the first position and then fill the rest 5 positions using letters G, I, R, T and U.
No. of such words = $5!$ (Since there are 5 distinct letters and 5 positions to be filled).
Now comes the word starting with G. As we need to find the serial number of the GUITAR, we need to check whether there is any other word coming in front of it. As after G, there comes U in the word GUITAR. As there are letters A, I, R and T before U as per the dictionary order, we will obtain other words starting with G coming before the word GUITAR.
Fixing G in the first position of the word and for the second position, let us choose the letter A. Then there will be four positions remaining and four letters.
Number of letters starting with G and with letter A on second = $4!$
Similarly fixing G at first and letters I, R, T for the second position, there will be $4!$ words for each case.
So in total, the number of words starting with G and letters A, I, R, T on the second position =$4! + 4! + 4! + 4!$
Now let us count the words which come before the word GUITAR with first letter G and second letter U. Fix G and U for the first and second positions and there will be A, I, T, R for the rest. A coming for the third place implies there are 3 positions left with three letters. Then the number of such words = $3!$ .
Now coming to the words with letters G, U, I in the order respectively. Then there are a few such words which can come before the word GUITAR which are GUIART, GUIATR, GUIRAT and GUIRTA. Thus there are four words before the word GUITAR starting with G, then U and then I.
Thus the total number of words coming before the word GUITAR as per dictionary order is $5! + (4! + 4! + 4! + 4!) + 3! + 4$
$ = 120 + (4 \times 24) + 6 + 4$
$ = 226$
Thus the word GUITAR will be on the serial number 227

So, the correct answer is “Option C”.

Note:Appropriate visualization of the word given will correctly lead us to the solution. As they have given a six letter word, we could imagine six vacant places with 6 objects (here it is letters) to fill. According to the conditions provided, we could calculate which places are to be filled with which objects.