In how many ways can 30 marks be allotted to 8 question if each question carries at least 2 marks?

Explanation

To solve the problem of distributing 30 marks among 8 questions, where each question receives at least 2 marks, we can follow these steps:

### Step 1: Allocate Minimum Marks
Since each question must receive at least 2 marks, we start by allocating 2 marks to each of the 8 questions.

\[ \text{Marks allocated} = 2 \times 8 = 16 \text{ marks} \]

### Step 2: Calculate Remaining Marks
After allocating the minimum marks, we calculate how many marks are left to distribute.

\[ \text{Remaining marks} = 30 - 16 = 14 \text{ marks} \]

### Step 3: Use the Stars and Bars Theorem
Now, we need to distribute the remaining 14 marks among the 8 questions. This is a combinatorial problem that can be solved using the "Stars and Bars" theorem.

In this case, we need to find the number of ways to distribute \( r \) indistinguishable objects (remaining marks) into \( n \) distinguishable boxes (questions). The formula is given by:

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

Where:
- \( n \) is the number of questions (8)
- \( r \) is the number of remaining marks (14)

### Step 4: Substitute Values into the Formula
Substituting the values into the formula:

\[ \binom{8 + 14 - 1}{14} = \binom{21}{14} \]

### Step 5: Calculate the Binomial Coefficient
The binomial coefficient can be calculated as follows:

\[ \binom{21}{14} = \frac{21!}{14! \cdot (21 - 14)!} = \frac{21!}{14! \cdot 7!} \]

### Step 6: Simplify the Factorial Expression
Now we can simplify this expression:

\[ \binom{21}{14} = \frac{21 \times 20 \times 19 \times 18 \times 17 \times 16 \times 15}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

Calculating the numerator:

\[ 21 \times 20 = 420 \]
\[ 420 \times 19 = 7980 \]
\[ 7980 \times 18 = 143640 \]
\[ 143640 \times 17 = 2431680 \]
\[ 2431680 \times 16 = 38906880 \]
\[ 38906880 \times 15 = 583603200 \]

Calculating the denominator:

\[ 7! = 5040 \]

Now, divide the numerator by the denominator:

\[ \frac{583603200}{5040} = 115584 \]

### Final Answer
Thus, the total number of ways to allot 30 marks to 8 questions, with each question receiving at least 2 marks, is:

\[ \boxed{115584} \]