Let P be a set of 21 integers from - 10 to 10, i.e. ` P = { -10,-9,-8-7…..7,8,9,10}` In how many ways can one select 19 integers from the above set such that their sum comes to one ?

Answer

To solve the problem, we need to find the number of ways to select 19 integers from the set \( P = \{-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) such that their sum equals 1.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a total of 21 integers in the set \( P \). To select 19 integers, we can equivalently think of this as ignoring 2 integers from the set.

2. **Calculating the Total Sum**:
If we include all integers from -10 to 10, the sum will be:
\[ \text{Sum} = (-10) + (-9) + ... + 0 + ... + 9 + 10 = 0 \]
This is because every negative integer cancels out with its positive counterpart.

3. **Setting Up the Condition**:
We need the sum of the selected 19 integers to equal 1. Since the total sum is 0, ignoring two integers \( x \) and \( y \) will give us:
\[ 0 - (x + y) = 1 \implies x + y = -1 \]

4. **Finding Pairs of Integers**:
We need to find pairs \( (x, y) \) such that \( x + y = -1 \) and both \( x \) and \( y \) belong to the set \( P \).

5. **Identifying Valid Pairs**:
Let's list the pairs:
- If \( x = 0 \), then \( y = -1 \) (pair: (0, -1))
- If \( x = 1 \), then \( y = -2 \) (pair: (1, -2))
- If \( x = 2 \), then \( y = -3 \) (pair: (2, -3))
- If \( x = 3 \), then \( y = -4 \) (pair: (3, -4))
- If \( x = 4 \), then \( y = -5 \) (pair: (4, -5))
- If \( x = 5 \), then \( y = -6 \) (pair: (5, -6))
- If \( x = 6 \), then \( y = -7 \) (pair: (6, -7))
- If \( x = 7 \), then \( y = -8 \) (pair: (7, -8))
- If \( x = 8 \), then \( y = -9 \) (pair: (8, -9))
- If \( x = 9 \), then \( y = -10 \) (pair: (9, -10))

6. **Counting the Valid Pairs**:
We have identified 10 valid pairs:
- (0, -1)
- (1, -2)
- (2, -3)
- (3, -4)
- (4, -5)
- (5, -6)
- (6, -7)
- (7, -8)
- (8, -9)
- (9, -10)

7. **Conclusion**:
Therefore, the number of ways to select 19 integers from the set \( P \) such that their sum equals 1 is **10**.