Question

Using all the letters of the word MATHS, then rank of the word THAMS is:

• A. 101
• B. 102
• C. 103
• D. 104

Answer 1

The Correct Option is C

Given:

We are tasked to determine the rank of the word "THAMS" in alphabetical order using the factorial method.

Step 1: Assign Numerical Positions to Letters

The alphabetical order of the letters T, H, A, M, S is:

\[ A = 1, \, H = 2, \, M = 3, \, S = 4, \, T = 5. \]

Thus, the word "THAMS" corresponds to the sequence: \( 5, 2, 1, 3, 4 \).

Step 2: Count Permutations for Each Letter

• For \( T \): There are 4 letters after \( T \) in alphabetical order (\( H, A, M, S \)). The number of permutations is: \[ 4! = 24. \]
• For \( H \): After \( H \), there is 1 letter (\( A \)) that comes earlier in the sequence. The number of permutations is: \[ 3! \cdot 1 = 6. \]
• For \( A \): There are no letters after \( A \) that need to be considered. The number of permutations is: \[ 2! \cdot 0 = 0. \]
• For \( M \): Similarly, there are no letters after \( M \) to consider. The number of permutations is: \[ 1! \cdot 0 = 0. \]
• For \( S \): There are no letters after \( S \). The number of permutations is: \[ 0! \cdot 0 = 0. \]

Step 3: Compute Total Permutations Before "THAMS"

Total permutations before "THAMS" is given by:

\[ 4 \cdot 4! + 3! \cdot 1 + 0 + 0 + 0 = 4 \cdot 24 + 6 = 96 + 6 = 102. \]

Step 4: Add 1 for the Word Itself

Rank of "THAMS" is:

\[ 102 + 1 = 103. \]

Final Answer:

The rank of the word "THAMS" is 103.