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 receives all the toys, we can follow these steps:

### Step 1: Understand the Problem
We have 15 identical toys and 15 students. The condition is that no single student can receive all 15 toys.

### Step 2: Use the Stars and Bars Theorem
The problem can be approached using the "stars and bars" theorem, which helps in distributing indistinguishable objects (toys) into distinguishable boxes (students). The formula for distributing \( n \) identical items into \( r \) distinct groups is given by:

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

In our case, \( n = 15 \) (toys) and \( r = 15 \) (students).

### Step 3: Calculate Total Distributions (Including Invalid Cases)
Using the formula, we calculate the total number of ways to distribute the toys without any restrictions:

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

### Step 4: Subtract Invalid Cases
Now, we need to subtract the cases where at least one student receives all 15 toys. If one student receives all 15 toys, the remaining 14 students receive none. There are 15 students who could potentially receive all the toys.

Thus, the number of invalid distributions (where one student gets all the toys) is 15.

### Step 5: Calculate Valid Distributions
Now, we can find the valid distributions by subtracting the invalid cases from the total distributions:

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

### Final Answer
The number of ways to distribute 15 identical toys among 15 students such that no student gets all the toys is:

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