The number of ways can five people be divided into three groups is

Options

A. `(a) 20`
B. (b) `25`
C. `(c) 30`
D. (d) `36`

Explanation

To solve the problem of dividing 5 people into 3 groups, we need to consider the possible distributions of people among the groups. The two feasible distributions for 5 people into 3 groups are:

1. One group with 3 people and two groups with 1 person each (1, 1, 3).
2. One group with 2 people and two groups with 2 people each (2, 2, 1).

We will calculate the number of ways for each distribution and then sum them up.

### Step 1: Calculate the number of ways for the distribution (1, 1, 3)

For this distribution, we can choose 3 people out of 5 to form the group of 3. The remaining 2 people will each form their own group.

- The number of ways to choose 3 people from 5 is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).

\[ C(5, 3) = \frac{5!}{3! \cdot (5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \]

Since the two groups of 1 person each are indistinguishable, we do not need to multiply by any additional factor.

### Step 2: Calculate the number of ways for the distribution (2, 2, 1)

For this distribution, we first choose 1 person to form the group of 1. The remaining 4 people will be divided into two groups of 2.

- The number of ways to choose 1 person from 5 is:

\[ C(5, 1) = \frac{5!}{1! \cdot (5-1)!} = 5 \]

- Next, we need to divide the remaining 4 people into 2 groups of 2. The number of ways to do this is given by:

\[ \frac{C(4, 2)}{2!} = \frac{\frac{4!}{2! \cdot 2!}}{2} = \frac{6}{2} = 3 \]

The division by \(2!\) accounts for the fact that the two groups of 2 are indistinguishable.

### Step 3: Combine the results

Now we can sum the number of ways from both distributions:

- For the distribution (1, 1, 3): 10 ways
- For the distribution (2, 2, 1): \(5 \times 3 = 15\) ways

Total number of ways:

\[ 10 + 15 = 25 \]

### Final Answer

The total number of ways to divide 5 people into 3 groups is **25**.

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