Priyanka has 11 different toys and supriya has 8 different roys . Find the number of ways in which they can exchange their toys so that each keeps her initial number of toys .

Options

A. `""^(19)C_(11)`
B. `""^(18)C_(10)`
C. `""^(20)C_(11)`
D. `""^(19)C_(11)-1`

Answer

To solve the problem of how many ways Priyanka and Supriya can exchange their toys while keeping their initial number of toys, we can break down the solution into clear steps.

### Step-by-Step Solution:

1. **Identify the Total Number of Toys**:
- Priyanka has 11 different toys.
- Supriya has 8 different toys.
- Therefore, the total number of toys combined is:
\[ 11 + 8 = 19 \]

2. **Understanding the Exchange**:
- When Priyanka and Supriya exchange toys, they must each end up with the same number of toys they started with. This means Priyanka will still have 11 toys and Supriya will still have 8 toys after the exchange.

3. **Choosing Toys for Exchange**:
- To maintain their original counts, Priyanka can choose any 11 toys from the total of 19 toys. The number of ways to choose 11 toys from 19 is given by the combination formula:
\[ \binom{19}{11} \]

4. **Calculating the Combinations**:
- The number of combinations can be calculated using the formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- In this case, we need to calculate:
\[ \binom{19}{11} = \frac{19!}{11! \cdot (19-11)!} = \frac{19!}{11! \cdot 8!} \]

5. **Accounting for the Case of No Exchange**:
- If they do not exchange any toys at all, that is a single case we need to subtract from our total. This is because the problem requires that at least one toy must be exchanged.
- Therefore, we subtract 1 from the total combinations:
\[ \text{Total ways} = \binom{19}{11} - 1 \]

6. **Final Answer**:
- The final answer is the number of ways they can exchange their toys while keeping their initial counts:
\[ \text{Final Answer} = \binom{19}{11} - 1 \]