There are 210 boys attending the national cricket academy at Bangalore. Each of them is assigned to work with one of the twenty coaches. It is noted that each coach works with a distinct distinct number of boys. In how many ways can the groups of boys be assigned?

Answer

To solve the problem of assigning 210 boys to 20 coaches, where each coach works with a distinct number of boys, we can follow these steps:

### Step 1: Understand the problem
We have 210 boys and 20 coaches. Each coach must have a different number of boys assigned to them, and the total number of boys assigned to all coaches must equal 210.

### Step 2: Assign distinct numbers of boys to each coach
Since each coach works with a distinct number of boys, we can denote the number of boys assigned to the coaches as \( x_1, x_2, \ldots, x_{20} \), where \( x_i \) is the number of boys assigned to the \( i \)-th coach. The condition we have is:
\[ x_1 + x_2 + \ldots + x_{20} = 210 \]
and \( x_1, x_2, \ldots, x_{20} \) must all be distinct positive integers.

### Step 3: Determine the smallest distinct integers
The smallest 20 distinct positive integers are 1, 2, 3, ..., 20. The sum of these integers can be calculated using the formula for the sum of the first \( n \) integers:
\[ S = \frac{n(n + 1)}{2} \]
For \( n = 20 \):
\[ S = \frac{20 \times 21}{2} = 210 \]
This means that the smallest distinct integers that can be assigned to the coaches already sum up to 210.

### Step 4: Assign the boys to the coaches
Now that we know the distinct integers that can be assigned to the coaches, we can assign the boys. The number of ways to assign 210 boys to these groups can be calculated using the multinomial coefficient.

The number of ways to assign boys to the coaches is given by:
\[ \frac{210!}{1! \times 2! \times 3! \times \ldots \times 20!} \]

### Final Answer
Thus, the total number of ways to assign the groups of boys to the coaches is:
\[ \frac{210!}{1! \times 2! \times 3! \times \ldots \times 20!} \]

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