If all the permutations of the word STACK are arranged in dictionary order, then what will be the rank of the wordSTACK

1. Break Down the Problem

To determine the rank of the word "STACK" when all permutations are arranged in dictionary order, we need to consider how many words come before "STACK" in this arrangement.

2. Relevant Concepts

• The word "STACK" has 5 distinct letters: S, T, A, C, K.
• We will find all permutations that precede "STACK" by fixing the first letter and considering the arrangements of the remaining letters.

3. Analysis and Detail

1. List of letters in alphabetical order:

   • A, C, K, S, T

2. Finding words that start with each letter before S:

   • Starting with A: The permutations of (C, K, S, T) $4! = 24 \text{ permutations}$
   • Starting with C: The permutations of (A, K, S, T) $4! = 24 \text{ permutations}$
   • Starting with K: The permutations of (A, C, S, T) $4! = 24 \text{ permutations}$

3. Words starting with S: We can now consider permutations that start with S and the second letter:

   • The remaining letters are A, C, K, T, in alphabetical order: A, C, K, T.

4. Finding words starting with SA: This will lead to permutations of (C, K, T): $3! = 6 \text{ permutations}$

5. Finding words starting with SC: This will lead to permutations of (A, K, T): $3! = 6 \text{ permutations}$

6. Finding words starting with SK: This will lead to permutations of (A, C, T): $3! = 6 \text{ permutations}$

7. Finally, words starting with ST:

   • Next letter is A, and we then consider permutations of (C, K): $2! = 2 \text{ permutations} \text{ (these are "STACK" and "STA CK")}$
   • Thus "STACK" is the first permutation among these.

4. Verify and Summarize

Now, we can add up all the permutations calculated:

• Permutations starting with A: 24
• Permutations starting with C: 24
• Permutations starting with K: 24
• Permutations starting with SA: 6
• Permutations starting with SC: 6
• Permutations starting with SK: 6
• Permutations starting with ST (before STACK): 2

Total permutations before "STACK": $24 + 24 + 24 + 6 + 6 + 6 + 2 = 92$

Final Answer

The rank of the word "STACK" is $92 + 1 = 93$. Thus, the rank of "STACK" is 93.