Question

how many words can be made using alphabets of king. Find rank of KING in all the possible words made my scrambling it without repetition

Answer 1

The number of words that can be made using the alphabets of KING is 24. The rank of KING in all the possible words made by scrambling it without repetition is 16.

Answer 2

Let the given word be KING. The letters in the word are K, I, N, G. There are 4 distinct letters.

Step 1: Find the total number of words that can be made by scrambling the letters of KING without repetition. Since there are 4 distinct letters, the total number of words that can be formed is the number of permutations of 4 distinct objects, which is 4!. Total number of words = $4! = 4 \times 3 \times 2 \times 1 = 24$.

Step 2: Find the rank of the word KING in the lexicographically sorted list of all possible words. First, arrange the letters of the word KING in alphabetical order: G, I, K, N. The word is KING. We count the number of words that come before KING in the alphabetical order.

Consider the letters from left to right in the word KING:

1. First letter is K. The letters in alphabetical order are G, I, K, N. Letters smaller than K are G and I (2 letters). Words starting with G: The remaining 3 letters (I, K, N) can be arranged in $3!$ ways. Number of words = $3! = 6$. Words starting with I: The remaining 3 letters (G, K, N) can be arranged in $3!$ ways. Number of words = $3! = 6$. Number of words starting with a letter smaller than K = $6 + 6 = 12$.

2. Second letter is I. We are considering words starting with K. The remaining letters are I, N, G. Arrange these remaining letters in alphabetical order: G, I, N. The second letter of KING is I. The letters among {G, I, N} that are smaller than I is G (1 letter). Words starting with KG: The remaining 2 letters (I, N) can be arranged in $2!$ ways. Number of words = $2! = 2$. Number of words starting with KI and having a second letter smaller than I = 2.

3. Third letter is N. We are considering words starting with KI. The remaining letters are N, G. Arrange these remaining letters in alphabetical order: G, N. The third letter of KING is N. The letters among {G, N} that are smaller than N is G (1 letter). Words starting with KIG: The remaining 1 letter (N) can be arranged in $1!$ way. Number of words = $1! = 1$. Number of words starting with KIN and having a third letter smaller than N = 1.

4. Fourth letter is G. We are considering words starting with KIN. The remaining letter is G. Arrange this remaining letter in alphabetical order: G. The fourth letter of KING is G. The letters among {G} that are smaller than G is none (0 letters). Words starting with KING and having a fourth letter smaller than G = 0.

The total number of words that come before KING in the alphabetical order is the sum of the counts from each step: Number of words before KING = (Words starting with G or I) + (Words starting with KG) + (Words starting with KIG) Number of words before KING = $12 + 2 + 1 = 15$.

The rank of the word KING is the number of words before it plus 1. Rank of KING = (Number of words before KING) + 1 = $15 + 1 = 16$.

The number of words that can be made using the alphabets of KING is 24. The rank of KING in the lexicographically sorted list of these words is 16.