The total number of ways of selecting 6 coins out of 10 one rupee coins, 6 fifty paise coins and 8 twenty paise coins is :

Options

A. 28
B. 14
C. 13
D. 19

Answer

To solve the problem of selecting 6 coins from the given denominations, we can use the concept of combinations and the stars and bars theorem. Let's break down the solution step by step.

### Step 1: Define the Variables
We have three types of coins:
- 1 Rupee coins: 10 available
- 50 Paise coins: 6 available
- 20 Paise coins: 8 available

Let:
- \( x_1 \) = number of 1 Rupee coins selected
- \( x_2 \) = number of 50 Paise coins selected
- \( x_3 \) = number of 20 Paise coins selected

We need to find the total number of ways to select 6 coins, which can be represented by the equation:
\[ x_1 + x_2 + x_3 = 6 \]

### Step 2: Set Constraints
We also have constraints based on the available coins:
- \( 0 \leq x_1 \leq 10 \)
- \( 0 \leq x_2 \leq 6 \)
- \( 0 \leq x_3 \leq 8 \)

### Step 3: Use the Stars and Bars Method
The stars and bars theorem allows us to find the number of non-negative integer solutions to the equation \( x_1 + x_2 + x_3 = 6 \) without considering the upper limits initially.

The number of non-negative integer solutions is given by:
\[ \binom{n + k - 1}{k - 1} \]
where \( n \) is the total number of coins to select (6) and \( k \) is the number of types of coins (3).

So, we calculate:
\[ \binom{6 + 3 - 1}{3 - 1} = \binom{8}{2} \]

### Step 4: Calculate the Combination
Now we compute \( \binom{8}{2} \):
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

### Step 5: Verify Constraints
Now we check if the constraints affect our solution:
- The maximum for \( x_1 \) is 10, which is not a constraint since we are only selecting 6 coins.
- The maximum for \( x_2 \) is 6, which means we cannot select more than 6 of 50 Paise coins.
- The maximum for \( x_3 \) is 8, which is also not a constraint since we are only selecting 6 coins.

Since all constraints are satisfied, we conclude that our initial calculation holds.

### Final Answer
Thus, the total number of ways to select 6 coins from the given denominations is **28**.

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