Permutations

Since, each letter can be posted in any one of the five different letter boxes. So, a letter can be posted in 5 ways. Since, there are six letters and each can be posted in 5 ways. So, total number of ways

The number of faces is equal to the number of colours.
Therefore the number of ways of painting them is 1.

The number of words starting from A are 5! = 120
The number of words starting from I are 5! = 120
The number of words starting from KA are 4! = 24
The number of words starting from KI are 4! = 24
The number of words starting from KN are 4! = 24
The number of words starting from KRA are 3! = 6
The number of words starting from KRIA are 2! = 2
The number of words starting from KRIN are 2! = 2
The number of words starting from KRISA are 1! = 1
The number of words starting from KRISNA are 1! = 1

Hence, rank of word �KRISNA
= 2(120) + 3(24) + 6 + 2(2) + 2(1) = 324

Step 1: There are 12 vacancies to be filled, and 3 of these vacancies are reserved for candidates from the scheduled caste.
- The number of ways to select 3 candidates from the 5 scheduled caste candidates is .
- After selecting the 3 scheduled caste candidates, 9 vacancies remain, and they can be filled by any of the remaining 22 candidates. The number of ways to select 9 candidates from the 22 is .
Step 2: Therefore, the total number of ways to fill the vacancies is:

Step 1: Recall that combinations and permutations are given by the formulas: Thus, Step 2: Substitute these into the equation: Simplify: Step 3: Further simplifying the factorials: This gives: Step 4: Solve for : Using the quadratic formula: The positive root gives .

Step 1: Evaluating inverse trigonometric values.
Step 2: Summing up the values.

Step 1: Differentiating both sides.
Given: Differentiating both sides with respect to : Step 2: Using derivative formulas.
We use the derivative of inverse tangent: Since is also a function of , we apply the chain rule: Step 3: Rewriting the equation.
Rearranging, Thus, the correct answer is (C).