A contest consists of ranking `10` songs of which `6` are Indian classic and `4` are westorn songs. Number of ways of ranking so that (mention correct statements)

Options

A. There are exactly `3` indian classic songs in top `5` is `(5!)^(3)`.
B. Top rank goes to Indian classic song is `6(9!)`
C. The ranks of all western songs are consecutive is `4!7!`
D. The `6` Indian classic songs are in a specified order is `"^(10)P_(4)`.

Explanation

To solve the problem of ranking 10 songs, consisting of 6 Indian classic songs and 4 Western songs, we will analyze each option step by step.

### Step 1: Understanding the Problem
We have 10 songs in total:
- 6 Indian classic songs (let's denote them as I1, I2, I3, I4, I5, I6)
- 4 Western songs (let's denote them as W1, W2, W3, W4)

We need to find the number of ways to rank these songs under specific conditions provided in the options.

### Step 2: Analyzing Each Option

#### Option 1: Exactly 3 Indian classic songs in the top 5
1. **Choose 3 Indian songs from 6**: This can be done in \( \binom{6}{3} \) ways.
2. **Choose 2 Western songs from 4**: This can be done in \( \binom{4}{2} \) ways.
3. **Arrange the 5 chosen songs**: The arrangement of these 5 songs can be done in \( 5! \) ways.

The total number of ways for this option is:
\[ \text{Total} = \binom{6}{3} \times \binom{4}{2} \times 5! \]

Calculating:
- \( \binom{6}{3} = 20 \)
- \( \binom{4}{2} = 6 \)
- \( 5! = 120 \)

Thus,
\[ \text{Total} = 20 \times 6 \times 120 = 14400 \]

#### Option 2: Top rank goes to an Indian classic song
1. **Choose 1 Indian song for the top rank**: This can be done in \( \binom{6}{1} \) ways.
2. **Arrange the remaining 9 songs**: The arrangement of these 9 songs can be done in \( 9! \) ways.

The total number of ways for this option is:
\[ \text{Total} = \binom{6}{1} \times 9! \]

Calculating:
- \( \binom{6}{1} = 6 \)
- \( 9! = 362880 \)

Thus,
\[ \text{Total} = 6 \times 362880 = 2177280 \]

#### Option 3: The rank of all Western songs are consecutive
1. **Treat the 4 Western songs as a single unit**: This gives us 7 units to arrange (6 Indian songs + 1 Western unit).
2. **Arrange these 7 units**: This can be done in \( 7! \) ways.
3. **Arrange the 4 Western songs within their unit**: This can be done in \( 4! \) ways.

The total number of ways for this option is:
\[ \text{Total} = 7! \times 4! \]

Calculating:
- \( 7! = 5040 \)
- \( 4! = 24 \)

Thus,
\[ \text{Total} = 5040 \times 24 = 120960 \]

#### Option 4: The 6 Indian classic songs are in a specific order
1. **Arrange the 6 Indian songs in a specific order**: This can be done in \( 1 \) way (since they are in a specific order).
2. **Arrange the remaining 4 Western songs**: This can be done in \( 4! \) ways.
3. **Arrange all 10 songs**: This can be done in \( 10! \) ways, but we need to divide by the arrangements of the Indian songs since they are in a specific order.

The total number of ways for this option is:
\[ \text{Total} = \frac{10!}{4!} \]

Calculating:
- \( 10! = 3628800 \)
- \( 4! = 24 \)

Thus,
\[ \text{Total} = \frac{3628800}{24} = 151200 \]

### Final Summary
All four options are valid and have been calculated as follows:
1. **Option 1**: 14400 ways
2. **Option 2**: 2177280 ways
3. **Option 3**: 120960 ways
4. **Option 4**: 151200 ways