If p, q, r in N, then the number of points having position vector `phati+qhatj+rhatk` such that `8lep+q+r le12` are

Options

A. 110
B. 185
C. 144
D. 108

Answer

To solve the problem, we need to find the number of points with position vectors of the form \( \vec{r} = p\hat{i} + q\hat{j} + r\hat{k} \) such that \( 8 \leq p + q + r \leq 12 \), where \( p, q, r \) are natural numbers (i.e., \( p, q, r \in \mathbb{N} \)).

### Step-by-Step Solution:

1. **Understand the Range of the Sum**:
We need to find the number of solutions for the equation \( p + q + r = n \) where \( n \) can take values from 8 to 12.

2. **Convert to Non-Negative Integer Solutions**:
Since \( p, q, r \) are natural numbers, we can redefine them as \( p' = p - 1 \), \( q' = q - 1 \), \( r' = r - 1 \) where \( p', q', r' \geq 0 \). This means we need to solve:
\[ (p' + 1) + (q' + 1) + (r' + 1) = n \implies p' + q' + r' = n - 3 \]
Thus, we need to find solutions for \( p' + q' + r' = n - 3 \) for \( n = 8, 9, 10, 11, 12 \).

3. **Calculate for Each Value of n**:
- For \( n = 8 \):
\[ p' + q' + r' = 8 - 3 = 5 \]
The number of non-negative integer solutions is given by the formula \( \binom{n + r - 1}{r - 1} \):
\[ \text{Number of solutions} = \binom{5 + 3 - 1}{3 - 1} = \binom{7}{2} = 21 \]

- For \( n = 9 \):
\[ p' + q' + r' = 9 - 3 = 6 \]
\[ \text{Number of solutions} = \binom{6 + 3 - 1}{3 - 1} = \binom{8}{2} = 28 \]

- For \( n = 10 \):
\[ p' + q' + r' = 10 - 3 = 7 \]
\[ \text{Number of solutions} = \binom{7 + 3 - 1}{3 - 1} = \binom{9}{2} = 36 \]

- For \( n = 11 \):
\[ p' + q' + r' = 11 - 3 = 8 \]
\[ \text{Number of solutions} = \binom{8 + 3 - 1}{3 - 1} = \binom{10}{2} = 45 \]

- For \( n = 12 \):
\[ p' + q' + r' = 12 - 3 = 9 \]
\[ \text{Number of solutions} = \binom{9 + 3 - 1}{3 - 1} = \binom{11}{2} = 55 \]

4. **Total Solutions**:
Now, we sum up all the solutions:
\[ \text{Total} = 21 + 28 + 36 + 45 + 55 = 185 \]

### Final Answer:
The total number of points having the position vector \( p\hat{i} + q\hat{j} + r\hat{k} \) such that \( 8 \leq p + q + r \leq 12 \) is **185**.