The number of ways in which 12 students can be equally divided into three groups isa)5775b)7575c)7755d)none of theseCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

12C4 × 8C4 × 4C4
= 495 × 70×1 = 34650

34650 ÷ 3!
= 5775

Total ways = 5775

Answer

Answer

Given: 12 students need to be divided into three groups.

To find: Number of ways in which this can be done.

Method:

We can use the formula for finding the number of ways of partitioning n distinct objects into k non-empty subsets.

The formula is given by:

P(n, k) = 1/k! * (n! / (n1! * n2! * ... * nk!))

where n1, n2, ..., nk are the number of objects in each of the k subsets.

In this case, we have n = 12 and k = 3. We need to find the number of ways in which we can divide 12 students into 3 groups.

Let the number of students in each group be x, y, and z respectively.

Then, we have the following conditions:

x + y + z = 12 (since all 12 students need to be divided)

x, y, z > 0 (since each group should have at least one student)

We can solve this using the method of stars and bars.

We represent the students as stars and the divisions between groups as bars. We need to place two bars to divide the 12 stars into three groups.

For example, if we have 4 students in the first group, 5 in the second group, and 3 in the third group, we can represent it as:

****|*****|***

The two bars divide the stars into three groups.

Using this method, the number of ways in which we can divide 12 students into 3 groups is:

P(12, 3) = 1/3! * (12! / (x! * y! * z!))

We need to find the sum of all possible values of x, y, and z that satisfy the conditions.

We can do this by fixing the value of x and finding the number of ways in which we can divide the remaining students into two groups.

Let's try some values of x:

If x = 1, we have 11 students remaining. We can divide them into two groups in P(11, 2) ways.

If x = 2, we have 10 students remaining. We can divide them into two groups in P(10, 2) ways.

If x = 3, we have 9 students remaining. We can divide them into two groups in P(9, 2) ways.

...

If x = 6, we have 6 students remaining. We can divide them into two groups in P(6, 2) ways.

Note that we cannot have x > 6, since that would leave fewer than 3 students for one of the groups.

Therefore, the total number of ways in which we can divide 12 students into 3 groups is:

P(12, 3) = 1/3! * (11! / (1! * 5! * 5!)) + 1/3! * (10! / (2! * 4! * 4!)) + ... + 1/3! * (6! / (6! * 0! * 0!))

Simplifying this expression, we get:

P(12, 3) = 1/6 * (11*10*9)/(1*2) + 1/6 * (10*9*8)/(2*

Answer

We are asked to find the number of ways in which 12 students can be equally divided into three groups. This is a problem of distributing the students into three groups where the groups are indistinguishable, and each group has 4 students.

Step 1: Total ways to divide students into 3 groups
The number of ways to divide 12 students into 3 groups of 4 students each is given by the formula for partitioning n objects into r groups of equal size:

Ways = ¹²C₄ × ⁸C₄ × ⁴C₄ / 3!

Step 2: Calculation of combinations
First, we calculate the individual combinations:

¹²C₄ = (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1) = 495

⁸C₄ = (8 × 7 × 6 × 5) / (4 × 3 × 2 × 1) = 70

⁴C₄ = 1

Step 3: Apply the formula
Now, we apply the formula to find the total number of ways:

Ways = (495 × 70 × 1) / 6 = 5775

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