Question

The letters of the word ‘ZENITH’ are written in all possible orders. How many words are possible if all these words are written out as in a dictionary? What is the rank of the word ‘ZENITH’?

Answer 1

Hint: We will count how many letters are in the word ‘ZENITH’. Further we will write in the dictionary type and calculate the rank.


Complete step by step solution:
In ZENITH, it has $6$ letters with no repetition. Therefore the number of ways of arranging $6$position is
$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
$6! = 720$
Now, we will arrange ZENITH in the alphabetical order. So, E,I,H,N,T,Z
Now, we will calculate the process of word ZENITH rank in the dictionary.
As, we know that first word starting from the letter $5! = 5 \times 4 \times 3 \times 2 \times 1$
First word starting from the letter$ = 120$
Number of words starting with $I = 5!$
Number of words starting with $I$$ = 5 \times 4 \times 3 \times 2 \times 1$
Number of words starting with $I = 120$
In the same way
Number of words starting with $H = 5!$
Number of words starting with $H = 5 \times 4 \times 3 \times 2 \times 1$
Number of words starting with $H = 120$
Number of words starting with $T = 5!$
Number of words starting with $T = 5 \times 4 \times 3 \times 2 \times 1$
Number of words starting with $T = 120$
So, first we find the words being with ZEN,ZENI, ZENH
Now, words starting from ZEN$ = 3!$
Words starting from ZEN $ = 3 \times 2 \times 1$
Words starting from ZEN$ = 6$
Word starting from ZEI$ = 3!$
Words starting from ZEI $ = 3 \times 2 \times 1$
Words starting from ZEI$ = 6$
Words starting from ZENH$ = 2!$
Words starting from ZENH$ = 2 \times 1$
Words starting from ZENH$ = 2$
Words starting from ZENIH$ = 1!$
Words starting from ZENIH$ = 1$
Words starting from ZENITH$ = 1!$
Words starting from ZENITH$ = 1$
So, total number of intermediate words
$
   = 5 \times 120 + 6 + 6 + 2 + 1 + 1 \\
   = 600 + 12 + 4 \\
   = 616 \\
 $
Hence, the rank of the word ZENITH when arranged in the dictionary is $616$


Note: Student keep in mind that you will rewrite the ZENITH in the alphabetical order as per the letters occurring in the dictionary.Also,care should be taken about the repeating letters