The number of ways of distributing 15 identical toys among some or all of the students in a classroom consisting of 15 students such that no student gets all the toys

Options

A. `2^(15)-15`
B. `.^(29)C_(14)-15`
C. `.^(30)C_(15)-15`
D. `.^(29)C_(15)-1`

Answer

To solve the problem of distributing 15 identical toys among 15 students such that no student gets all the toys, we can follow these steps:

### Step 1: Understand the Total Distribution
We need to find the total number of ways to distribute 15 identical toys among 15 students without any restrictions. The formula for distributing \( n \) identical items to \( r \) distinct groups is given by:

\[ \binom{n + r - 1}{r - 1} \]

In our case, \( n = 15 \) (toys) and \( r = 15 \) (students). Thus, we can calculate:

\[ \text{Total ways} = \binom{15 + 15 - 1}{15 - 1} = \binom{29}{14} \]

### Step 2: Calculate the Case Where One Student Gets All Toys
Next, we need to account for the cases where at least one student receives all the toys. If one student receives all 15 toys, the remaining 14 students receive none. There are 15 students, so there are 15 ways to choose which student receives all the toys.

### Step 3: Subtract the Unwanted Cases
To find the number of ways to distribute the toys such that no student gets all the toys, we subtract the cases where one student gets all the toys from the total distribution calculated in Step 1.

\[ \text{Valid distributions} = \binom{29}{14} - 15 \]

### Step 4: Final Calculation
Now we can compute the final answer. The number of ways to distribute 15 identical toys among 15 students such that no student gets all the toys is:

\[ \text{Number of ways} = \binom{29}{14} - 15 \]