Question

A dictionary is printed consisting of 7 lettered words only that can be made with the letter of word CRICKET. If the words are printed at the alphabetical order, as in an ordinary dictionary, then the number of word before the word CRICKET is
$\left( a \right)$ 530
$\left( b \right)$ 480
$\left( c \right)$ 531
$\left( d \right)$ 481

Answer 1

Hint: In this particular question use the concept that first fixed the first and second place letters and arrange the remaining letters then fixed the first and third place letters at starting and arrange the remaining letters this process continues till we get the given word so use these concepts to reach the solution of the question.

Complete step by step answer:
Given word
CRICKET
As we see there are 7 letters in the given word.
Now we have to arrange them in a dictionary.
So first write the place value of these letters.
C – 1, C – 2, E – 3, I – 4, K – 5, R – 6, T – 7.
Now we have to arrange them in a dictionary.
So if the first and second letters are C so the number of ways to arrange the remaining 5 letters = 5!
Now if first C and second E so the number of ways to arrange the remaining 5 letters = 5!
Now if first C and second I so the number of ways to arrange the remaining 5 letters = 5!
Now if first C and second K so the number of ways to arrange the remaining 5 letters = 5!
Now if first C, second R and third C so the number of ways to arrange the remaining 4 letters = 4!
Now if first C, second R and third E so the number of ways to arrange the remaining 4 letters = 4!
Now if first C, second R, third I, fourth C, fifth E so the number of ways to arrange the remaining 2 letters = 2!
Now if first C, second R, third I, fourth C, fifth K, sixth E and seventh T = 1 way
So the position of word CRICKET in the dictionary is the sum of all the above cases.
Therefore, the position of the word CRICKET in the dictionary is = $4 \times 5! + 2 \times 4! + 2! + 1$
Now simplify it we have,
Therefore, position of word CRICKET in the dictionary is
= $4 \times \left( {120} \right) + 2 \times \left( {24} \right) + 2 + 1 = 480 + 48 + 3 = 531$
So the number of words before the given word CRICKET = 531 – 1 = 530.
So this is the required answer.

So, the correct answer is “Option A”.

Note: Whenever we face such types of questions the key concept we have to remember is that the position of word CRICKET in the dictionary is the sum of all the cases as above and always recall that the number of ways to arrange n different objects are n!, so after addition just simplify and subtract the last word from the summation value we will get the required number of words before the given word CRICKET.