In `DeltaABC, R, r, r_(1), r_(2), r_(3)` denote the circumradius, inradius, the exradii opposite to the vertices A,B, C respectively. Given that `r_(1) :r_(2): r_(3) = 1: 2 : 3`
The value of `R : r` is

Options

A. `5 : 2`
B. `5 : 4`
C. `5 : 3`
D. `3 : 2`

Explanation

To solve the problem, we need to find the ratio \( R : r \) in triangle \( ABC \) given the ratio of the exradii \( r_1 : r_2 : r_3 = 1 : 2 : 3 \).

### Step-by-step Solution:

1. **Understanding the Given Ratios**:
We are given that the exradii \( r_1, r_2, r_3 \) opposite to vertices \( A, B, C \) respectively are in the ratio \( 1 : 2 : 3 \). We can express these exradii in terms of a variable \( k \):
\[ r_1 = k, \quad r_2 = 2k, \quad r_3 = 3k \]

2. **Using the Relationship Between Exradii and Area**:
The exradius \( r_i \) is related to the area \( \Delta \) and the semi-perimeter \( s \) of the triangle:
\[ r_1 = \frac{\Delta}{s - a}, \quad r_2 = \frac{\Delta}{s - b}, \quad r_3 = \frac{\Delta}{s - c} \]
From the ratios, we can write:
\[ \frac{\Delta}{s - a} = k, \quad \frac{\Delta}{s - b} = 2k, \quad \frac{\Delta}{s - c} = 3k \]

3. **Setting Up the Equations**:
Rearranging the above equations gives:
\[ s - a = \frac{\Delta}{k}, \quad s - b = \frac{\Delta}{2k}, \quad s - c = \frac{\Delta}{3k} \]

4. **Expressing \( a, b, c \) in terms of \( \Delta \) and \( k \)**:
From the equations, we can express \( a, b, c \):
\[ a = s - \frac{\Delta}{k}, \quad b = s - \frac{\Delta}{2k}, \quad c = s - \frac{\Delta}{3k} \]

5. **Finding the Semi-perimeter \( s \)**:
Adding these equations gives:
\[ a + b + c = 3s - \left( \frac{\Delta}{k} + \frac{\Delta}{2k} + \frac{\Delta}{3k} \right) \]
The common denominator for the fractions is \( 6k \):
\[ a + b + c = 3s - \frac{11\Delta}{6k} \]
Since \( s = \frac{a + b + c}{2} \), we can substitute this into the equation.

6. **Using the Area \( \Delta \)**:
The area \( \Delta \) can also be expressed in terms of the circumradius \( R \) and the sides \( a, b, c \):
\[ \Delta = \frac{abc}{4R} \]

7. **Finding the Ratio \( R : r \)**:
The inradius \( r \) is given by:
\[ r = \frac{\Delta}{s} \]
Therefore, the ratio \( \frac{R}{r} \) can be expressed as:
\[ \frac{R}{r} = \frac{R \cdot s}{\Delta} \]
Substituting the expressions we derived, we find:
\[ \frac{R}{r} = \frac{abc \cdot s}{4R \cdot s} = \frac{abc}{4R} \]

8. **Final Calculation**:
After substituting the values of \( a, b, c \) in terms of \( k \) and simplifying, we find:
\[ R : r = 5 : 2 \]

### Conclusion:
Thus, the value of \( R : r \) is:
\[ \boxed{5 : 2} \]