If the maximum number of trials required to open all locks when there are n locks and n keys is 105, then n =

Options

A. 13
B. 14
C. 35
D. 20

Answer

To solve the problem, we need to find the value of \( n \) given that the maximum number of trials required to open all locks when there are \( n \) locks and \( n \) keys is 105.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have \( n \) locks and \( n \) keys.
- The maximum number of trials needed to open all locks is given as 105.

2. **Calculating Maximum Trials**:
- For the first key, we have \( n \) trials (since it can try to open any of the \( n \) locks).
- For the second key, we have \( n - 1 \) trials (it can only try to open \( n - 1 \) locks if the first key has already opened one).
- Continuing this pattern, for the \( r \)-th key, the number of trials needed is \( n - r + 1 \).

3. **Summing the Trials**:
- The total number of trials needed can be expressed as:
\[ T = n + (n - 1) + (n - 2) + \ldots + 1 \]
- This is the sum of the first \( n \) natural numbers, which can be calculated using the formula:
\[ T = \frac{n(n + 1)}{2} \]

4. **Setting Up the Equation**:
- We know that the total trials equal 105:
\[ \frac{n(n + 1)}{2} = 105 \]

5. **Solving for \( n \)**:
- Multiply both sides by 2 to eliminate the fraction:
\[ n(n + 1) = 210 \]
- Rearranging gives us a quadratic equation:
\[ n^2 + n - 210 = 0 \]

6. **Applying the Quadratic Formula**:
- The quadratic formula is given by:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 1 \), \( b = 1 \), and \( c = -210 \):
\[ n = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-210)}}{2 \cdot 1} \]
\[ n = \frac{-1 \pm \sqrt{1 + 840}}{2} \]
\[ n = \frac{-1 \pm \sqrt{841}}{2} \]
\[ n = \frac{-1 \pm 29}{2} \]
- This gives us two possible solutions:
\[ n = \frac{28}{2} = 14 \quad \text{and} \quad n = \frac{-30}{2} = -15 \]
- Since \( n \) must be a positive integer, we take \( n = 14 \).

7. **Conclusion**:
- The value of \( n \) is \( 14 \).