Three persons have 4 coats, 5 waistcoats ,and 6 hats. Find in how many ways can they put on the clothes.

Answer

To solve the problem of how many ways three persons can put on 4 coats, 5 waistcoats, and 6 hats, we will use the concept of permutations. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the problem
We have 3 persons and each person can choose from a different set of clothing items: 4 coats, 5 waistcoats, and 6 hats. We need to find out how many different combinations of clothing can be worn by these 3 persons.

### Step 2: Calculate the number of ways to choose coats
The number of ways in which 3 persons can choose from 4 coats is given by the permutation formula \( nPr \), where \( n \) is the total number of items and \( r \) is the number of items to choose. Here, we need to choose 3 coats from 4:

\[ 4P3 = \frac{4!}{(4-3)!} = \frac{4!}{1!} = 4 \times 3 \times 2 = 24 \text{ ways} \]

### Step 3: Calculate the number of ways to choose waistcoats
Next, we calculate the number of ways in which 3 persons can choose from 5 waistcoats:

\[ 5P3 = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 5 \times 4 \times 3 = 60 \text{ ways} \]

### Step 4: Calculate the number of ways to choose hats
Now, we calculate the number of ways in which 3 persons can choose from 6 hats:

\[ 6P3 = \frac{6!}{(6-3)!} = \frac{6!}{3!} = 6 \times 5 \times 4 = 120 \text{ ways} \]

### Step 5: Calculate the total number of ways
To find the total number of ways the three persons can wear the coats, waistcoats, and hats, we multiply the number of ways for each clothing item:

\[ \text{Total ways} = 4P3 \times 5P3 \times 6P3 = 24 \times 60 \times 120 \]

Calculating this gives:

\[ 24 \times 60 = 1440 \]
\[ 1440 \times 120 = 172800 \]

### Final Answer
Thus, the total number of ways in which the three persons can put on the clothes is **172800 ways**.

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