Permutations

It can be obtained by Starting from 1 = 5! = 120

We are tasked with finding the number of four-digit numbers that satisfy the following conditions:

• The number must be between 5000 and 10000.
• The digits must come from the set .
• No digit may be repeated.

The first digit must ensure the number is between 5000 and 10000. Therefore, it must be , , or . This gives us 3 choices for the first digit.

Once the first digit is chosen, there are 4 remaining digits in the set (excluding the chosen first digit). The second digit can then be any one of these 4 digits, giving 4 choices for the second digit. For the third digit, we are left with 3 choices (the remaining digits after the first two have been selected). Similarly, for the fourth digit, there are 2 remaining choices.

The total number of valid numbers is given by the product of choices for each position:

Thus, there are 72 four-digit numbers that satisfy the given conditions.

The correct answer is (A) : 16800
The word has 12 letters, out of which 5 are vowels and 7 are consonants.
Because we need to keep the vowels together always, we can consider the 5 vowels as one letter.
So, the number of ways of arranging the consonants is given by,
On expanding the factorial, we get ,
On simplification,
=3360
So the vowels can also be rearranged themselves. Out of the 5 vowels, 4 are the same.
So, the ways of arranging the vowels is given by
After simplification, we get,
Therefore, the vowels can be arranged in 5 ways.
The number of words that can be formed such that vowels are always together is given by the product of the number of ways of arranging the letters with all the vowels together and the number of ways of arranging the vowels.

The correct answer is (A) :
First, we need to find the total number of ways to seat all 12 people around the circular table, which is (12-1)! = 11! since we can fix one person's position as a reference.
Next, we need to subtract the number of ways that two or more girls sit together. We can approach this by treating the five girls as a block and permuting them first, which can be done in 5! ways.
Then we can insert this block of girls in the 8 spaces between the 7 boys or at the beginning or end of the line of boys, which gives us 9 positions to place the block of girls. Once the block of girls is placed, we can permute the 7 boys in 7! ways. Therefore, the total number of ways that two or more girls sit together is 5! × 9 × 7!
the number of ways that no two girls sit together is 11! - 5! × 9 × 7! = 126(5!)².

The number of four-digit numbers that can be formed using the digits 2, 1, 2, and 3 is .
These are the permutations of the digits 2, 1, 2, and 3. The sum of digits at the unit place is calculated as: Now, the required sum is: Thus, the sum is .

To solve the problem, we need to find the number of ways to arrange 3 girls and 4 boys such that all the girls are together, all the boys are together, and and are not adjacent. Let's solve this step-by-step:

1. Consider the girls and boys as a single block: The girls form one block and the boys form another block. We can arrange these 2 blocks in ways.

2. Arrangement within each block:

• Within the girl block, the 3 girls can be arranged in ways.
• The 4 boys can be arranged in ways.

3. Accounting for and not adjacent: First, calculate the total arrangement of boys, then the restriction.

• Total arrangements of boys is .
• If and are treated as a single unit, the arrangement within boys is .
• Inside the "unit" , they can switch places in ways, giving a total arrangement of .

The number of arrangements where and are together is , so there are arrangements where and are not adjacent.

4. Total arrangements not adjacent: Now, total combinations where and are not adjacent: .

5. Apply probability that boys without and adjacent: Since the unwanted arrangement was added twice, double count requires division by 2 not needed as one group of boys and one for , total valid arrangements stands corrected at ways when double counted correctly as part of boys block overlap error.

This means the total number of arrangements where all conditions are met is 72.

Step 1: Arrange letters alphabetically.
Alphabetical order: Step 2: Count permutations before each letter.
Compute permutations starting with letters smaller than the current letter at each position.
Step 3: Sum all preceding permutations.
Total number of words before ``UDAYPUR'' is Step 4: Add 1 for the word itself.