Permutations And Combinations

The correct answer is C:
Given expression;

Since, the first 2 women select the chairs amongst 1 to 4 in ways. Now, from the remaining 6 chairs, three men could be arranged in
Total number of arrangements =

Since, a five-digit number is formed using the digits and divisible by i.e. only possible when sum of the digits is multiple of three.
Case I Using digits
Number of ways
Case II Using digits
Number of ways
Total numbers formed

Since, .3.5

Total number of divisors
Out of these 2, 6, 10, and 30 are of the form 4n + 2.
Therefore, (a) is the answer.

Distinct n-digit numbers which can be formed using digits 2,5 and 7 are 3 We have to find n, so that
3

n 7, so the least value of n is 7

Let n be the number of newspapers which are read by the students.
Then,

Since, r, s, t are prime numbers,
Selection of p and q are as under
p q Number of ways
1 way
1 way
3 way
Total number of ways to select, r = 5
Selection of s as under
1 way
1 way
1 way
1 way
5 way
Total number of ways to select s = 9
Similarly, the number of ways to select t = 5
Total number of ways = 5 x 9 x 5 = 225

Arrange the letters of the word COCHIN as in the order
of dictionary CCHINO.
Consider the words starting from C
.
There are 5! such words. Number of words with the two
C's occupying first and second place = 4!.
Number of words starting with CH, Cl, CN is 4! each
.
Similarly, number of words before the first word
starting with CO = 4! + 4! + 4! + 4! = 96.
The word starting with CO found first in the dictionary
is COCHIN. There are 96 words before COCHIN.

Step 1: Understand the set

- is the set of all seven-digit numbers formed using the digits 0, 1, and 2.

- A seven-digit number has the form , where each .

- Since it’s a seven-digit number, the first digit cannot be 0 (otherwise, it wouldn’t be a seven-digit number; e.g., 0210222 would be interpreted as 210222, a six-digit number).

- Thus:

\qquad - (2 choices),

\qquad - (3 choices each).

- Total number of seven-digit numbers in :
- Compute , so:
- Therefore, .
The example confirms this:

- 2210222 is a seven-digit number with digits from \{0, 1, 2\}, so it’s in .

- 0210222 is not a seven-digit number (it’s 210222, a six-digit number), so it’s not in . Step 2: Define the condition

- We need to find the number of elements such that the number of digits 0 and the number of digits 1 in are equal.

- Let:

\qquad - Number of 0s in ,

\qquad - Number of 1s in ,

\qquad - Number of 2s in .

- Since the total number of digits is 7, and the number of 2s must be non-negative:
- Also, . So, can be 0, 1, 2, or 3. Step 3: Count the numbers for each

For each value of , compute the number of valid seven-digit numbers, considering the restriction on the first digit.
Case 1:

- 0s: 0, 1s: 0, 2s: .

- All digits are 2: the number is 2222222.

- First digit is 2, which is fine.

- Number of such numbers: 1.
Case 2:

- 0s: 1, 1s: 1, 2s: .

- Total digits: 7.

- Choose 1 position out of 7 for the 0: .

- From the remaining 6 positions, choose 1 for the 1: .

- The remaining 5 positions are 2s.

- Total ways (without considering the first digit): .

- Now, exclude numbers where the first digit is 0:

\qquad - First digit is 0: Fix position 1 as 0 (1 way).

\qquad - Choose 1 position out of the remaining 6 for the 1: .

\qquad - Remaining 5 positions are 2s.

\qquad - Number of invalid cases: 6.

- Valid cases: .
Case 3:

- 0s: 2, 1s: 2, 2s: .

- Choose 2 positions out of 7 for the 0s: .

- From the remaining 5 positions, choose 2 for the 1s: .

- Remaining 3 positions are 2s.

- Total ways: .

- Exclude cases where the first digit is 0:

\qquad - First digit is 0: Fix position 1 as 0.

\qquad - Choose 1 more position out of positions 2 to 7 for the other 0: .

\qquad - From the remaining 5 positions, choose 2 for the 1s: .

\qquad - Remaining 3 positions are 2s.

\qquad - Invalid cases: .

- Valid cases: .
Case 4:

- 0s: 3, 1s: 3, 2s: .

- Choose 3 positions out of 7 for the 0s: .

- From the remaining 4 positions, choose 3 for the 1s: .

- Remaining 1 position is a 2.

- Total ways: .

- Exclude cases where the first digit is 0:

\qquad - First digit is 0: Fix position 1 as 0.

\qquad - Choose 2 more positions out of positions 2 to 7 for the other 0s: .

\qquad - From the remaining 4 positions, choose 3 for the 1s: .

\qquad - Remaining position is a 2.

\qquad - Invalid cases: .

- Valid cases: .
Step 5: Sum the valid cases

- : 1,

- : 36,

- : 150,

- : 80.

- Total: Final Answer: The number of elements where the number of 0s equals the number of 1s is .