There are 10 girls and 8 boys in a class room including Mr. Ravi, Ms. Rani and Ms. Radha. A list of speakers consisting of 8 girls and 6 boys has to be prepared. Mr. Ravi refuses to speak if Ms.Rani is a speaker. Ms. Rani refuses to speak if Ms. Radha is a speaker. The number of ways the list can be prepared is a 3 digit number `n_1n_2n_3`, then `|n_3 + n_2 - n_1|`

Answer

To solve the problem step by step, we will analyze the conditions and calculate the number of ways to prepare the list of speakers.

### Step 1: Understand the Problem
We have 10 girls and 8 boys in total, including Mr. Ravi, Ms. Rani, and Ms. Radha. We need to prepare a list of 8 girls and 6 boys. However, there are restrictions based on who can speak.

### Step 2: Identify Cases Based on Conditions
We will consider three cases based on the selections of Ms. Rani and Ms. Radha.

1. **Case 1: Ms. Rani is selected.**
- If Ms. Rani is selected, Mr. Ravi cannot be selected.
- We have already selected 1 girl (Ms. Rani), so we need to select 7 more girls from the remaining 9 girls.
- The number of ways to select 7 girls from 9 is given by \( \binom{9}{7} \).
- For boys, we need to select 6 from the 7 remaining boys (since Mr. Ravi cannot be selected).
- The number of ways to select 6 boys from 7 is given by \( \binom{7}{6} \).
- Therefore, the total for this case is:
\[ \text{Case 1 Total} = \binom{9}{7} \times \binom{7}{6} = 36 \times 7 = 252. \]

2. **Case 2: Ms. Radha is selected.**
- If Ms. Radha is selected, Ms. Rani cannot be selected.
- We need to select 7 girls from the remaining 9 girls (since Ms. Rani is not included).
- The number of ways to select 7 girls from 9 is \( \binom{9}{7} \).
- For boys, we can select 6 boys from the 8 available boys.
- The number of ways to select 6 boys from 8 is \( \binom{8}{6} \).
- Therefore, the total for this case is:
\[ \text{Case 2 Total} = \binom{9}{7} \times \binom{8}{6} = 36 \times 28 = 1008. \]

3. **Case 3: Neither Ms. Rani nor Ms. Radha is selected.**
- We need to select 8 girls from the remaining 8 girls (since neither Ms. Rani nor Ms. Radha is included).
- The number of ways to select 8 girls from 8 is \( \binom{8}{8} = 1 \).
- For boys, we need to select 6 boys from the 8 available boys.
- The number of ways to select 6 boys from 8 is \( \binom{8}{6} \).
- Therefore, the total for this case is:
\[ \text{Case 3 Total} = \binom{8}{8} \times \binom{8}{6} = 1 \times 28 = 28. \]

### Step 3: Calculate Total Ways
Now, we sum the totals from all three cases:
\[ \text{Total Ways} = \text{Case 1 Total} + \text{Case 2 Total} + \text{Case 3 Total} = 252 + 1008 + 28 = 1288. \]

### Step 4: Extract the Digits
The total number of ways is 1288. We need to extract the digits:
- \( n_1 = 1 \)
- \( n_2 = 2 \)
- \( n_3 = 8 \)

### Step 5: Calculate the Final Expression
We need to compute \( |n_3 + n_2 - n_1| \):
\[ |n_3 + n_2 - n_1| = |8 + 2 - 1| = |9| = 9. \]

### Final Answer
The final answer is \( 9 \).

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