Question 1192321
Certainly, let's break down these geometry problems:

**1. Number of Lines from n Points**

* **Concept:** To form a line, we need to select 2 points from the given 'n' points. 
* **Calculation:** The number of ways to select 2 points from 'n' points is given by the combination formula: 
    * nC2 = n! / (2! * (n-2)!) 
    * where 'n!' denotes the factorial of n (n! = n * (n-1) * (n-2) * ... * 1)

**Therefore, the number of lines that can be formed from 'n' non-collinear points is nC2.**

**2. Number of Planes from n Points**

* **Concept:** To form a plane, we need to select 3 non-collinear points.
* **Calculation:** The number of ways to select 3 points from 'n' points is given by: 
    * nC3 = n! / (3! * (n-3)!)

**Therefore, the number of planes that can be formed from 'n' non-coplanar points is nC3.**

**3. Proof of Theorem 1.1.4**

* **a. A ∩ B is a point p.** 
    * **Reason:** By definition, intersecting lines share exactly one point in common.

* **b. B contains a point q ≠ p.**
    * **Reason:** Since B is a line, it must contain at least two distinct points. 

* **c. There is a plane E, containing A and q.**
    * **Reason:** Two distinct points (p and q) determine a unique line. A line and a point not on that line uniquely determine a plane.

* **d. E contains A ∪ B.**
    * **Reason:** 
        * E contains line A (by definition).
        * E contains point q (by definition).
        * Since E contains a point (p) and a line (A) that lies entirely within the plane, it must also contain all other points on that line (including all points on line B).

* **e. No other plane contains A ∪ B.**
    * **Reason:** Two intersecting lines uniquely define a plane. 

I hope this explanation is helpful! Let me know if you have any further questions.