4 five-rupee coins, 3 two-rupee coins and 2 one-rupee coins are stacked together in a column at random. The probability that the coins of the same denomination are consecutive is

Options

A. `13//9!`
B. `1//210`
C. `1//35`
D. none of these

Answer

To solve the problem, we need to find the probability that coins of the same denomination are consecutive when we stack 4 five-rupee coins, 3 two-rupee coins, and 2 one-rupee coins together in a column at random.

### Step 1: Calculate the total number of coins
We have:
- 4 five-rupee coins
- 3 two-rupee coins
- 2 one-rupee coins

Total number of coins = 4 + 3 + 2 = 9 coins.

### Step 2: Calculate the total number of arrangements of the coins
The total arrangements of these coins can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \]
where:
- \( n \) is the total number of items (9 coins),
- \( n_1 \) is the number of five-rupee coins (4),
- \( n_2 \) is the number of two-rupee coins (3),
- \( n_3 \) is the number of one-rupee coins (2).

Thus, we have:
\[ \text{Total arrangements} = \frac{9!}{4! \cdot 3! \cdot 2!} \]

### Step 3: Calculate the number of arrangements where coins of the same denomination are together
To ensure that coins of the same denomination are together, we can treat each denomination as a single unit or block. Therefore, we have:
- 1 block of 5-rupee coins (4 coins),
- 1 block of 2-rupee coins (3 coins),
- 1 block of 1-rupee coins (2 coins).

This gives us 3 blocks in total. The arrangements of these blocks can be calculated as:
\[ \text{Arrangements of blocks} = 3! = 6 \]

Within each block, the coins can be arranged among themselves:
- The 4 five-rupee coins can be arranged in \( 4! \) ways.
- The 3 two-rupee coins can be arranged in \( 3! \) ways.
- The 2 one-rupee coins can be arranged in \( 2! \) ways.

Thus, the total arrangements where coins of the same denomination are together is:
\[ \text{Total arrangements with blocks} = 3! \cdot 4! \cdot 3! \cdot 2! \]

### Step 4: Calculate the probability
Now, we can find the probability \( P \) that the coins of the same denomination are consecutive:
\[ P = \frac{\text{Number of arrangements where coins are together}}{\text{Total arrangements}} \]
Substituting the values we calculated:
\[ P = \frac{3! \cdot 4! \cdot 3! \cdot 2!}{\frac{9!}{4! \cdot 3! \cdot 2!}} \]

### Step 5: Simplifying the probability expression
\[ P = \frac{3! \cdot 4! \cdot 3! \cdot 2! \cdot 4! \cdot 3! \cdot 2!}{9!} \]
This simplifies to:
\[ P = \frac{6 \cdot 24 \cdot 6 \cdot 2}{362880} = \frac{288}{362880} = \frac{1}{1260} \]

### Final Answer
Thus, the probability that the coins of the same denomination are consecutive is:
\[ \frac{1}{210} \]