The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is

Options

A. `2^(20)-1`
B. `2^(21)`
C. `2^(20)`
D. `2^(20)+1`

Answer

To solve the problem of choosing 10 objects from a total of 31 objects, where 10 are identical and 21 are distinct, we can break down the solution step by step.

### Step 1: Understand the Composition of Objects
We have:
- 10 identical objects (let's call them A)
- 21 distinct objects (let's call them B1, B2, ..., B21)

### Step 2: Define the Selection Scenarios
When choosing 10 objects, we can choose a certain number of identical objects (A) and the rest from the distinct objects (B). Let \( x \) be the number of identical objects chosen. Therefore, \( x \) can take values from 0 to 10 (i.e., \( x = 0, 1, 2, ..., 10 \)). The remaining objects will then be selected from the distinct objects.

### Step 3: Calculate the Number of Distinct Objects Chosen
If we choose \( x \) identical objects, we need to choose \( 10 - x \) distinct objects from the 21 available distinct objects. The number of ways to choose \( 10 - x \) distinct objects from 21 is given by the binomial coefficient \( \binom{21}{10 - x} \).

### Step 4: Sum Over All Possible Values of \( x \)
We need to sum the number of ways for all possible values of \( x \):
\[ S = \sum_{x=0}^{10} \binom{21}{10 - x} \]
This can be rewritten as:
\[ S = \binom{21}{10} + \binom{21}{9} + \binom{21}{8} + \ldots + \binom{21}{0} \]

### Step 5: Use the Identity for Binomial Coefficients
Using the identity for binomial coefficients, we know that:
\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \]
In our case, we can use this identity to find:
\[ \sum_{k=0}^{21} \binom{21}{k} = 2^{21} \]
However, since we are only summing from \( k = 0 \) to \( k = 10 \), we can use the symmetry of binomial coefficients:
\[ \sum_{k=0}^{10} \binom{21}{k} = \sum_{k=11}^{21} \binom{21}{k} \]
Thus, we have:
\[ \sum_{k=0}^{10} \binom{21}{k} = \frac{1}{2} \cdot 2^{21} = 2^{20} \]

### Final Result
Therefore, the total number of ways to choose 10 objects from the given set is:
\[ S = 2^{20} \]

### Conclusion
The answer is \( 2^{20} \).

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