To solve the problem of distributing 21 identical blankets among three beggars such that exactly two of them receive an equal number of blankets, we can follow these steps:
### Step 1: Define Variables
Let the number of blankets received by the first beggar be \( x \), the second beggar also receive \( x \), and the third beggar receive \( y \). Thus, we can express the total number of blankets as:
\[
2x + y = 21
\]
### Step 2: Rearrange the Equation
From the equation \( 2x + y = 21 \), we can rearrange it to find \( y \):
\[
y = 21 - 2x
\]
### Step 3: Determine Constraints
Since \( y \) must be non-negative (the beggar cannot receive a negative number of blankets), we have:
\[
21 - 2x \geq 0
\]
This simplifies to:
\[
2x \leq 21 \quad \Rightarrow \quad x \leq 10.5
\]
Since \( x \) must be a non-negative integer, the maximum value for \( x \) is 10.
### Step 4: Find Possible Values of \( x \)
The values of \( x \) can be integers from 0 to 10. However, we also need to ensure that \( y \) is non-negative and that \( y \) must be an odd integer (since \( y \) must be odd for \( 2x + y \) to equal 21 while \( 2x \) is even).
### Step 5: Determine Values of \( y \)
Substituting values of \( x \) from 0 to 10 into \( y = 21 - 2x \):
- If \( x = 0 \), then \( y = 21 - 2(0) = 21 \) (valid)
- If \( x = 1 \), then \( y = 21 - 2(1) = 19 \) (valid)
- If \( x = 2 \), then \( y = 21 - 2(2) = 17 \) (valid)
- If \( x = 3 \), then \( y = 21 - 2(3) = 15 \) (valid)
- If \( x = 4 \), then \( y = 21 - 2(4) = 13 \) (valid)
- If \( x = 5 \), then \( y = 21 - 2(5) = 11 \) (valid)
- If \( x = 6 \), then \( y = 21 - 2(6) = 9 \) (valid)
- If \( x = 7 \), then \( y = 21 - 2(7) = 7 \) (valid)
- If \( x = 8 \), then \( y = 21 - 2(8) = 5 \) (valid)
- If \( x = 9 \), then \( y = 21 - 2(9) = 3 \) (valid)
- If \( x = 10 \), then \( y = 21 - 2(10) = 1 \) (valid)
### Step 6: Count Valid Cases
Now, we need to count how many of these values of \( y \) are odd. The valid pairs \((x, y)\) are:
- \( (0, 21) \)
- \( (1, 19) \)
- \( (2, 17) \)
- \( (3, 15) \)
- \( (4, 13) \)
- \( (5, 11) \)
- \( (6, 9) \)
- \( (7, 7) \)
- \( (8, 5) \)
- \( (9, 3) \)
- \( (10, 1) \)
All values of \( y \) are odd, and there are 11 valid cases.
### Step 7: Conclusion
Thus, the total number of ways to distribute the 21 identical blankets among the three beggars such that exactly two of them receive the same number of blankets is:
\[
\boxed{11}
\]
