If 10 coins of Rs. 10, five coins of Rs. 5 are to be placed in a line, then the probability that the extreme coins are of Rs. 5 is

Options

A. `(1)/(15!)`
B. `(1)/(10!)`
C. `(5!10!)/(15!)`
D. None of these

Answer

To solve the problem of finding the probability that the extreme coins are of Rs. 5 when placing 10 coins of Rs. 10 and 5 coins of Rs. 5 in a line, we can follow these steps:

### Step 1: Determine the total number of coins
We have:
- 10 coins of Rs. 10
- 5 coins of Rs. 5

Total coins = 10 + 5 = 15 coins.

### Step 2: Calculate the total arrangements of the coins
The total arrangements of these 15 coins can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{15!}{10! \times 5!} \]

### Step 3: Fix the extreme coins
To find the favorable outcomes where both extreme coins are Rs. 5, we fix two Rs. 5 coins at the ends. This leaves us with:
- 10 coins of Rs. 10
- 3 coins of Rs. 5 (since we used 2 Rs. 5 coins)

Now we have a total of:
10 (Rs. 10) + 3 (Rs. 5) = 13 coins left to arrange in the middle.

### Step 4: Calculate the arrangements of the remaining coins
The arrangements of these 13 coins can be calculated as:
\[ \text{Favorable arrangements} = \frac{13!}{10! \times 3!} \]

### Step 5: Calculate the probability
The probability that the extreme coins are Rs. 5 is given by the ratio of favorable arrangements to total arrangements:
\[ P(\text{extreme coins are Rs. 5}) = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{\frac{13!}{10! \times 3!}}{\frac{15!}{10! \times 5!}} \]

### Step 6: Simplify the probability expression
This simplifies to:
\[ P = \frac{13! \times 5!}{15! \times 3!} \]
Now, substituting \(15! = 15 \times 14 \times 13!\):
\[ P = \frac{5!}{15 \times 14 \times 3!} \]

### Step 7: Calculate the factorials
Calculating the factorials:
- \(5! = 120\)
- \(3! = 6\)

Substituting these values:
\[ P = \frac{120}{15 \times 14 \times 6} = \frac{120}{1260} = \frac{2}{21} \]

### Final Answer
Thus, the probability that the extreme coins are of Rs. 5 is:
\[ \frac{2}{21} \]