Arrange the following combinations in the increasing order of focal length
Two piano convex lenses of focal lengths 15 cm and 30 cm in contact
Two convex lens of focal lengths 40 cm and 50 cm in contact
Two convex lenses of focal length 20 cm separated by 5 cm

Options

A. a, b, c
B. b, a, c
C. a, c, b
D. c, a, b

Explanation

To solve the problem of arranging the combinations of lenses in increasing order of their focal lengths, we will use the lens formula for two lenses in contact and for two lenses separated by a distance.

### Step-by-Step Solution:

1. **Identify the combinations:**
- Combination A: Two piano convex lenses with focal lengths \( f_1 = 15 \, \text{cm} \) and \( f_2 = 30 \, \text{cm} \) in contact.
- Combination B: Two convex lenses with focal lengths \( f_1 = 40 \, \text{cm} \) and \( f_2 = 50 \, \text{cm} \) in contact.
- Combination C: Two convex lenses with focal length \( f_1 = 20 \, \text{cm} \) separated by a distance of \( d = 5 \, \text{cm} \).

2. **Calculate the focal length for Combination A:**
- For two lenses in contact, the formula for the equivalent focal length \( f \) is given by:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \]
- Substituting the values:
\[ \frac{1}{f} = \frac{1}{15} + \frac{1}{30} \]
- Finding a common denominator (30):
\[ \frac{1}{f} = \frac{2}{30} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10} \]
- Therefore, \( f = 10 \, \text{cm} \).

3. **Calculate the focal length for Combination B:**
- Using the same formula:
\[ \frac{1}{f} = \frac{1}{40} + \frac{1}{50} \]
- Finding a common denominator (200):
\[ \frac{1}{f} = \frac{5}{200} + \frac{4}{200} = \frac{9}{200} \]
- Therefore, \( f = \frac{200}{9} \approx 22.22 \, \text{cm} \).

4. **Calculate the focal length for Combination C:**
- For two lenses separated by a distance \( d \), the formula is:
\[ f = \frac{f_1 f_2}{f_1 + f_2 - d} \]
- Substituting the values:
\[ f = \frac{20 \times 20}{20 + 20 - 5} = \frac{400}{35} \approx 11.43 \, \text{cm} \]

5. **Arrange the focal lengths in increasing order:**
- Combination A: \( 10 \, \text{cm} \)
- Combination C: \( 11.43 \, \text{cm} \)
- Combination B: \( 22.22 \, \text{cm} \)

Thus, the increasing order of focal lengths is:
\[ A < C < B \]

### Final Answer:
The combinations in increasing order of focal length are:
- Combination A (10 cm)
- Combination C (11.43 cm)
- Combination B (22.22 cm)