Question

The letters of 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

Answer:

$\large\boxed{\sf{Total\:Possible\:Words= 720}}$

$\large\boxed{\sf{Rank\:of\:ZENITH=616}}$

Step-by-step explanation:

In the given word ZENITH, there are total 6 letters.

.°. Total possible words = 6! = 720

Now, to find the rank of ZENITH in dictionary.

• First of all write the numbers from 1 to 6 on the top of letters which comes in dictionary first to last.
• Then, from left to right, see the rank of the letter at top and write the number of letters which comes before that letter in dictionary.
• For example, in the word ZENITH, Z is numbered 6 on top as it comes last in dictionary, and 5 is written at its bottom because after Z , 5 letters are there which actually come before Z in dictionary.
• Similarly, for E, 1 is written on top and 0 at bottom because no letters after E in the word ZENITH comes before E in dictionary.
• Similarly, all letters are numbered accordingly.
• Now, from right to left , write 0! , 2!, 3! ......and so on at the bottom of each letter.
• Now, multiply each numbers in the bottom of letters and add them.

Therefore, We will get

= (5 × 5!) + (0 × 4!) + (2 × 3!) + (1 × 2!) + (1 × 1!) + (0 × 0!)

= 600 + 0 + 12 + 2 + 1 + 0

= 615

• Now, add 1 to the resultant sum. This will be the rank of the word.

Hence, rank of ZENITH = (615 + 1) = 616

$\bold\red{Note:-}$

• Refer to the attachment for concept.

Answer 2

Answer:

Number of words starting with E = 5! = 120

Number of words starting with H = 5! = 120

Number of words starting with I = 5! = 120

Number of words starting with N = 5! = 120

Number of words starting with T = 5! = 120

Now, the word will start with the letter Z.

After Z, alphabetically, the next letter would be E, which is as per the requirement of the word ZENITH.