Number System

Step 1: Prime factorize 6300

Step 2: Total divisors
Total number of positive divisors:
Step 3: Count only even divisors
Even divisors must include at least one factor of 2. So remove all divisors that do not include 2: Thus:

The given digits are 4, 6, 9, 5, 3, , and .
These are distinct digits, and since they must be distinct from the given five digits.
Also .
For a number to be divisible by 3, the sum of its digits must be divisible by 3.
Sum of the given five digits = .
The sum of all seven digits is .
For this sum to be divisible by 3, must be divisible by 3.
The possible pairs of distinct digits from such that is divisible by 3 are: \begin{itemize} \item (3 not available) \item (6 not available) \item \item (5 not available) \item \item (4 not available) \item \item \item (0, 3, 6, 9 - 3, 6, 9 not available) \end{itemize} The pairs from such that is divisible by 3 are: \begin{itemize} \item - not possible \item , sum = 3 \item , sum = 9 \item , sum = 3 \item , sum = 9 \item , sum = 9 \item , sum = 15 \item , sum = 9 \item , sum = 15 \end{itemize} The ordered pairs are: .
There are 8 such ordered pairs.

We need to find the highest power of 2 that divides .
We can use Legendre's formula for this: The exponent of a prime in the prime factorization of is given by: $ and .
\) $ So, the highest power of 2 that divides is .
We are given that divides and does not divide .
This means that is the exponent of the highest power of 2 that divides .
Therefore, .

The binomial coefficients for are: The product . Using known formula or induction: Hence, option (4) is correct.

The given expression is . We can rewrite this expression using negative exponents: To use the binomial expansion , we factor out 2 from the term : This is in the form , where , , and . The general formula for the term in the binomial expansion of is . For the term, we set , which implies . So, the term of is . First, calculate the binomial coefficient : This fraction can be simplified by dividing numerator and denominator by 3: . Next, calculate the term : . Now, combine these to find : . The term of the original expression is : . Finally, substitute the given value : . . To match the format of the options, we can factor : . Comparing this with the given options, specifically option (B) , we notice a discrepancy in the denominators. If our calculated result were to equal option (B), then must be equal to . However, and . Since , the exact calculated value does not numerically match option (B). Assuming Option (B) is the intended correct answer as indicated: The calculated numerical value of the 5th term is .

Step 1: Divisibility Rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9. Step 2: Digits used: From 0 to 9 (i.e., 10 digits total), choose 8 **distinct** digits such that their sum is divisible by 9. Total sum of digits 0 through 9: We are to select 8 digits out of these 10 such that their sum is divisible by 9. So, we need to **remove 2 digits** whose sum is divisible by 9. Step 3: Count valid digit sets: We find all pairs such that or (since total = 45, removing 9, 18, or 27 keeps the remaining sum divisible by 9): - : possible pairs = (0,9), (1,8), (2,7), (3,6), (4,5) 5 pairs - : (9, 9) invalid (can't repeat), and other valid unique pairs: (8,10) invalid, (7,11) invalid etc. So only valid pairs that actually exist within 0-9 = (9,9) [but repeating not allowed] 0 pairs - : check combinations within 0–9 whose sum is 27 and distinct (9, 8, 7, ..., 0): only (9, 8, 1), (9, 7, 2), etc. would be 3 digits. So only 5 valid 2-digit removal sets where sum = 9 So, total such pairs: **5** Step 4: For each of those 5 valid digit sets: Number of 8-digit numbers = permutations of those 8 digits But since first digit can't be 0, subtract cases where 0 is leading. Total ways for each set: Total ways over 5 sets: Factorized form: % Tip

Let S be the number of sisters, so . Let B be the number of brothers, so . Total number of people . The total number of ways to arrange distinct people around a circle is . So, the total number of ways to arrange all 11 people around a circle without any restrictions is . Now, we need to find the number of ways such that all three sisters are NOT seated together. This can be found by subtracting the number of ways where all three sisters ARE seated together from the total number of arrangements. Case: All three sisters are seated together. Treat the 3 sisters as a single block or unit. Now we have 1 unit (of sisters) + 8 brothers = 9 units. Arrange these 9 units around a circle. The number of ways to do this is . Within the block of 3 sisters, they can arrange themselves in ways. So, the number of ways where all three sisters are seated together is . Number of ways where all three sisters are NOT seated together = (Total arrangements) - (Arrangements where all three sisters are together) Let's check the options: % Option (A) % Option (B) % Option (C) % Option (D) Our calculated answer matches option (D). Alternative Method: Gap Method First, arrange the 8 brothers around a circle. The number of ways to do this is . When 8 brothers are arranged around a circle, there are 8 gaps between them. For the 3 sisters to not be seated together, they must be placed in these gaps, with at most one sister per gap. We have 8 gaps and we need to choose 3 of them for the sisters, which can be done in ways (permutation because the sisters are distinct). . So, the number of ways is . Let's convert this to the format of the options: . This doesn't seem to directly match. Let's convert to : . This doesn't seem to simplify easily to the target format. Let's re-evaluate : . This result is , which is not . Let's recheck the "all three sisters are not seated together" interpretation. "All three sisters are not seated together" typically means no two sisters are adjacent, OR it can mean that the block of three sisters is not together. The phrasing "all the three sisters are not seated together" usually implies that the block of 3 sisters is broken, but not necessarily that no two sisters are adjacent. However, in competitive exams, this phrase often means "no two sisters are together" if the context implies maximum separation. If it means "the block of 3 sisters is not formed", then the subtraction method is correct. Let's assume "all three sisters are not seated together" means that no two sisters are adjacent. Using the gap method, arrange the 8 brothers in a circle in ways. This creates 8 gaps between the brothers. To ensure no two sisters are together, place each of the 3 sisters in a distinct gap. This can be done in ways. So, the number of ways = . . Let's calculate : . These are different. The phrasing "all the three sisters are not seated together" means that it is not the case that (Sister Sister Sister). It does not mean (Sister NOT Sister Sister NOT Sister). Therefore, the subtraction method is correct. Total arrangements Arrangements where the 3 sisters are together Arrangements where the 3 sisters are NOT together This matches option (D). The ambiguity of the phrase "not together" is important here. Usually, "not together" means "at least one is separate" from the group. If it means "no two adjacent", the question would usually be phrased as "no two sisters sit together".

Step 1: We are given that divides .
So, we must find the highest power of 10 that divides .
Step 2: Note that:
So, we need to find the highest power of both 2 and 5 that divide .
Step 3: Check divisibility by 2 and 5:
Let’s calculate :
So, divisible by 2.
Step 4: Check modulo 4:
So, divisible by .
Step 5: Check divisibility by 5:
Since , :
So divisible by 5.
Step 6: Check divisibility by 25:
Now try modulo 25:
Euler’s theorem: Since , if
We can use successive powers or verify:
So:
Hence, the maximum power of 5 is , and of 2 is .

Let S be the number of sisters, so . Let B be the number of brothers, so . Total number of people . The total number of ways to arrange distinct people around a circle is . So, the total number of ways to arrange all 11 people around a circle without any restrictions is . Now, we need to find the number of ways such that all three sisters are NOT seated together. This can be found by subtracting the number of ways where all three sisters ARE seated together from the total number of arrangements. Case: All three sisters are seated together. Treat the 3 sisters as a single block or unit. Now we have 1 unit (of sisters) + 8 brothers = 9 units. Arrange these 9 units around a circle. The number of ways to do this is . Within the block of 3 sisters, they can arrange themselves in ways. So, the number of ways where all three sisters are seated together is . Number of ways where all three sisters are NOT seated together = (Total arrangements) - (Arrangements where all three sisters are together) Let's check the options: % Option (A) % Option (B) % Option (C) % Option (D) Our calculated answer matches option (D). Alternative Method: Gap Method First, arrange the 8 brothers around a circle. The number of ways to do this is . When 8 brothers are arranged around a circle, there are 8 gaps between them. For the 3 sisters to not be seated together, they must be placed in these gaps, with at most one sister per gap. We have 8 gaps and we need to choose 3 of them for the sisters, which can be done in ways (permutation because the sisters are distinct). . So, the number of ways is . Let's convert this to the format of the options: . This doesn't seem to directly match. Let's convert to : . This doesn't seem to simplify easily to the target format. Let's re-evaluate : . This result is , which is not . Let's recheck the "all three sisters are not seated together" interpretation. "All three sisters are not seated together" typically means no two sisters are adjacent, OR it can mean that the block of three sisters is not together. The phrasing "all the three sisters are not seated together" usually implies that the block of 3 sisters is broken, but not necessarily that no two sisters are adjacent. However, in competitive exams, this phrase often means "no two sisters are together" if the context implies maximum separation. If it means "the block of 3 sisters is not formed", then the subtraction method is correct. Let's assume "all three sisters are not seated together" means that no two sisters are adjacent. Using the gap method, arrange the 8 brothers in a circle in ways. This creates 8 gaps between the brothers. To ensure no two sisters are together, place each of the 3 sisters in a distinct gap. This can be done in ways. So, the number of ways = . . Let's calculate : . These are different. The phrasing "all the three sisters are not seated together" means that it is not the case that (Sister Sister Sister). It does not mean (Sister NOT Sister Sister NOT Sister). Therefore, the subtraction method is correct. Total arrangements Arrangements where the 3 sisters are together Arrangements where the 3 sisters are NOT together This matches option (D). The ambiguity of the phrase "not together" is important here. Usually, "not together" means "at least one is separate" from the group. If it means "no two adjacent", the question would usually be phrased as "no two sisters sit together".

Step 1: Compute the value of :
Step 2: Prime factorisation of 5040:
Step 3: Use the formula for total number of divisors of a number:
If , then the number of divisors of is:
Step 4: Apply the formula:

The word COMBINATION has 11 letters with repetitions: .
Step 1: Fix at the first and at the last positions.
Step 2: The middle position (6th) cannot be a vowel. Vowels are (total 5 vowels).
Step 3: Consonants left for the middle position are (3 consonants). So, 3 choices for the middle letter.
Step 4: Remaining 8 letters are arranged in the remaining 8 positions. Since letters and are fixed at ends and middle is fixed, total arrangements for these 8 letters is .
Step 5: Total number of such words = choices for middle letter arrangements of remaining letters = (considering the repeated letters and fixed ).

Calculate , and .
Use binomial identities and algebraic simplifications:
is false (the actual value differs), is true, and is true.
Since in Reason (R) is false, (R) is false even though (A) is true.

Step 1: Use the formula for permutations .
The given equation is .
LHS: .
RHS: .

Step 2: Substitute these into the equation.

Step 3: Simplify the factorial expressions.
Assuming and , we can cancel them:
Step 4: Solve for .
Calculate the division: .
.
.
So, .
Thus, .

Step 5: Verify the conditions for permutations.
For : .
Also, .
For : .
Also, .
Also, from denominators of factorials: and .
The value satisfies all these conditions: , , , .
The values selected are and , which are valid.
This matches option (4).

Step 1: Use the formula for permutations .
The given equation is .
LHS: .
RHS: .

Step 2: Substitute these into the equation.

Step 3: Simplify the factorial expressions.
Assuming and , we can cancel them:
Step 4: Solve for .
Calculate the division: .
.
.
So, .
Thus, .

Step 5: Verify the conditions for permutations.
For : .
Also, .
For : .
Also, .
Also, from denominators of factorials: and .
The value satisfies all these conditions: , , , .
The values selected are and , which are valid.
This matches option (4).

Step 1: Rewrite the expression
Step 2: Approximate numerator and denominator for large
Step 3: Express as product
Step 4: Use binomial expansion
Apply to both factors with: Calculate expansions and multiply. Step 5: Result