The letters of the word ‘CALCULUS’ are arranged as in a dictionary. What is the rank of word ‘CALCULUS’?
Answers
Step-by-step explanation:
Given word:
CALCULUS
Assign numerical values according to the order of alphabets:
⇒ A = 1
⇒ C, C = 2
⇒ L, L = 3
⇒ S = 4
⇒ U, U = 5
This can be written as:
⇒ C, C = 2
⇒ A = 1
⇒ L, L = 3
⇒ U, U = 5
⇒ S = 4
Or also as:
⇒ C = 2
⇒ A = 1
⇒ L = 3
⇒ C = 2
⇒ U = 5
⇒ L = 3
⇒ U = 5
⇒ S = 4
Now, we see:
1 number is less than 2 at its right side. Among C and the characters at its right side, no characters are repeating.
⇒ 1/2! × 2! × 2!
No numbers are less than 1 at its right side. Among A and the characters at its right side, no characters are repeating.
⇒ 0/2! × 2!
1 number is less than 3 at its right side. Among L and the characters at its right side, no characters are repeating.
⇒ 1/2! × 2!
No numbers are less than 2 at its right side. Among C and the characters at its right side, no characters are repeating.
⇒ 0/2!
2 numbers are less than 5 at its right side (3,4). Among U and the characters at its right side, no characters are repeating.
⇒ 2/2!
No numbers are less than 3 at its right side. Among L and the characters at its right side, no characters are repeating.
⇒ 0/0!
1 number is less than 5 at its right side. Among U and the characters at its right side, no characters are repeating.
⇒ 1/0!
No numbers are less than 4 at its right side. Among S and the characters at its right side, no characters are repeating.
⇒ 0/0!
Now, from bottom to up write 0!, 1!, 2!...
⇒ C = 2 ⇔ 7!
⇒ A = 1 ⇔ 6!
⇒ L = 3 ⇔ 5!
⇒ C = 2 ⇔ 4!
⇒ U = 5 ⇔ 3!
⇒ L = 3 ⇔ 2!
⇒ U = 5 ⇔ 1!
⇒ S = 4 ⇔ 0!
Multiply now:
⇒ 1/2! × 2! × 2! × 7!
- [∵ 7! = 1 × 2 × 3 × 4 × 5 × 6 × 7 = 5040]
⇒ 1/8 × 5040
⇒ 630
Similarly for all we get:
⇒ 630
⇒ 0
⇒ 30
⇒ 0
⇒ 6
⇒ 0
⇒ 1
⇒ 0
Finding their sum and adding 1:
⇒ 667 + 1
⇒ 668
∴ The rank of the word CALCULUS in dictionary is 668.