Permutations And Combinations

Step 1: Treat same-size books as groups. Treat each group of books as a single item. The number of ways to arrange the groups is (for the three groups). Within each group, the books can be arranged in , , and ways for large, medium, and small books respectively.
Step 2: Conclusion. Thus, the total number of ways is .

Step 1: Understand grouping.
We divide 10 students into:
- Team 1: 4 students
- Team 2: 3 students
- Team 3: 3 students
Step 2: Choose 4 students for the first team.

Step 3: From remaining 6 students choose 3 for second team.

Step 4: Remaining 3 automatically form third team.
So total arrangements:

Step 5: Divide by because two teams of 3 are identical.

Step 6: Compute values.


Final Answer:

Step 1: Understand the problem.
Each face of the cube can be painted in 6 different ways. Since there are 6 faces on the cube, the number of possible ways to paint the cube is 6!.
Step 2: Conclusion.
Thus, the number of ways is , as the arrangement of colours is fixed.
Final Answer:

Step 1: Understand the problem.
Each letter can be placed in one of the 5 boxes. Therefore, the total number of ways to post 6 letters is .
Step 2: Conclusion.
Thus, the correct number of ways is .
Final Answer:

Required numbers

if objects occupy the original places and none of the remaining objects occupies its original places then the number of such arrangements

Step 1: Understanding the problem.
The number of triangles that can be formed is given by selecting 3 points from the 3 sides. The calculation uses combinatorics and the number of ways to select points on each side.

Step 2: Conclusion.
The total number of triangles that can be formed is 205, corresponding to option (1).

The total number of distinct arrangements of the word KRISNA is , as the letter 'A' repeats twice. To find the rank, we calculate the number of words that precede KRISNA in the dictionary order, resulting in a rank of 341.
Step 2: Conclusion.
The correct answer is (B), 341.

This problem involves counting the number of ways to select three squares from a diagonal or parallel diagonal of a chessboard. Using combinatorics, the answer is .
Step 2: Conclusion.
The correct answer is (C), 252.

The total number of words without restriction is , and the number of words with no repetition is . Subtract the number of words without repetition from the total.
Final Answer:

Step 1: The total number of ways to distribute 3 prizes to 4 children, without restriction, is .
Step 2: We need to subtract the number of ways in which one child gets all three prizes. There are 4 ways for this to happen (one for each child).
Step 3: Thus, the total number of ways is .

Final Answer:

Required number of ways