The total number of combinations of 2n different things taken any one or more at a time and total number of combinations of n different things taken one or more at a time is in ratio `65:1` then the value of n is equal to

Options

A. 4
B. 5
C. 6
D. none of these

Answer

To solve the problem, we need to find the value of \( n \) given that the total number of combinations of \( 2n \) different things taken one or more at a time is in the ratio \( 65:1 \) with the total number of combinations of \( n \) different things taken one or more at a time.

### Step-by-Step Solution:

1. **Understanding Combinations**:
The total number of combinations of \( r \) items from \( n \) items is given by \( \binom{n}{r} \). The total number of combinations of \( n \) different things taken one or more at a time can be calculated as:
\[ \sum_{r=1}^{n} \binom{n}{r} = 2^n - 1 \]
This is because the total number of subsets of \( n \) items is \( 2^n \), and we subtract 1 to exclude the empty set.

2. **Calculating for \( 2n \)**:
Similarly, the total number of combinations of \( 2n \) different things taken one or more at a time is:
\[ \sum_{r=1}^{2n} \binom{2n}{r} = 2^{2n} - 1 \]

3. **Setting Up the Ratio**:
According to the problem, we have the ratio:
\[ \frac{2^{2n} - 1}{2^n - 1} = 65 \]

4. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ 2^{2n} - 1 = 65(2^n - 1) \]
Expanding the right side:
\[ 2^{2n} - 1 = 65 \cdot 2^n - 65 \]

5. **Rearranging the Equation**:
Rearranging the equation yields:
\[ 2^{2n} - 65 \cdot 2^n + 64 = 0 \]

6. **Substituting \( x = 2^n \)**:
Let \( x = 2^n \). The equation becomes:
\[ x^2 - 65x + 64 = 0 \]

7. **Using the Quadratic Formula**:
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -65, c = 64 \):
\[ x = \frac{65 \pm \sqrt{(-65)^2 - 4 \cdot 1 \cdot 64}}{2 \cdot 1} \]
\[ x = \frac{65 \pm \sqrt{4225 - 256}}{2} \]
\[ x = \frac{65 \pm \sqrt{3969}}{2} \]
\[ x = \frac{65 \pm 63}{2} \]

8. **Calculating the Roots**:
This gives us two possible values for \( x \):
\[ x = \frac{128}{2} = 64 \quad \text{and} \quad x = \frac{2}{2} = 1 \]

9. **Finding \( n \)**:
Since \( x = 2^n \):
- If \( x = 64 \), then \( 2^n = 64 \) implies \( n = 6 \).
- If \( x = 1 \), then \( 2^n = 1 \) implies \( n = 0 \).

However, since \( n \) must be a positive integer, we take \( n = 6 \).

### Final Answer:
The value of \( n \) is \( 6 \).