Permutation And Combination

Step 1: Understand the number formation.
We need to form 3-digit numbers using the digits from . The number must have three digits, and repetition is allowed.
Step 2: Analyze the options.
- The first digit of the 3-digit number cannot be 0, so it can be any one of the digits from . This gives us 8 possible choices for the first digit.
- The second and third digits can each be any of the 9 digits from , which gives us 9 possible choices for each of these digits.

Step 3: Calculate the total number of numbers.
The total number of 3-digit numbers is the product of the choices for each digit:
Step 4: Conclusion.
Thus, the number of 3-digit numbers that can be formed is , which corresponds to option (C).

Step 1: Identify possible cases.
We need to select 5 persons with at least 3 men in the committee. We can have the following cases: - Case 1: 3 men and 2 women - Case 2: 4 men and 1 woman - Case 3: 5 men and 0 women
Step 2: Calculate number of ways for each case.
- Case 1: Number of ways to select 3 men from 5 and 2 women from 7: - Case 2: Number of ways to select 4 men from 5 and 1 woman from 7: - Case 3: Number of ways to select 5 men from 5:
Step 3: Total number of ways.
Total = 210 + 35 + 1 = 246.
Step 4: Conclusion.
Thus, the number of ways to form the committee is 246, which corresponds to option (C).

Step 1: Treating books of each subject as a block.
Since the books of each subject are to be kept together, we can treat each subject as a single block. So, we have 3 blocks: one for History, one for Philosophy, and one for Economics.
Step 2: Arranging the blocks.
The number of ways to arrange these 3 blocks is .
Step 3: Arranging the books within each block.
- The 2 History books can be arranged in ways. - The 3 Philosophy books can be arranged in ways. - The 5 Economics books can be arranged in ways.
Step 4: Total number of arrangements.
The total number of arrangements is:
Step 5: Conclusion.
Therefore, the correct answer is (4) 8640.

Step 1: Treating vowels as a block.
The word "BUSINESS" contains the vowels U, I, and E. To ensure that the vowels always come together, we treat them as a single unit or block. So, we have the following letters left: B, S, N, S, and the block of vowels (UIE).
Step 2: Arranging the blocks.
Now, we have 6 units: B, S, N, S, and the vowel block. The number of ways to arrange these 6 units is , because the letter S is repeated twice.
Step 3: Arranging the vowels.
The 3 vowels (UIE) can be arranged among themselves in ways.
Step 4: Total number of arrangements.
The total number of ways to arrange the letters so that the vowels are together is:
Step 5: Conclusion.
Therefore, the correct answer is (3) 7!.