Part II.Work out problems. Show all the necessary steps Carefully.Each Question worths 3.5 points 1. A survey was conducted about the preferred mobile cell phone brand with accounting departme nt students as respondents and it was found out that 150 preferred Apple, 125 preferred Samsung, 100 preferred Techno, 50 preferred Apple and Samsung,40 preferred Apple and Techno, 35 preferred Samsung and Tckno, 20 preferred all three brands,Let the universal set U contain all the 300 accounting students who were surveyed. Find the number of students those preferred Apple only,Techno only, Samsung only and preferred non of the three brands.

Here's how to solve this problem using a Venn diagram and the principle of inclusion-exclusion:<br /><br />**1. Define the sets:**<br /><br />* **A:** Set of students who prefer Apple<br />* **S:** Set of students who prefer Samsung<br />* **T:** Set of students who prefer Techno<br /><br />**2. Write down the given information:**<br /><br />* |A| = 150<br />* |S| = 125<br />* |T| = 100<br />* |A ∩ S| = 50<br />* |A ∩ T| = 40<br />* |S ∩ T| = 35<br />* |A ∩ S ∩ T| = 20<br />* |U| = 300<br /><br />**3. Calculate the number of students who prefer only one brand:**<br /><br />* **Apple only:** |A| - (|A ∩ S| + |A ∩ T| - |A ∩ S ∩ T|) = 150 - (50 + 40 - 20) = 80<br />* **Samsung only:** |S| - (|A ∩ S| + |S ∩ T| - |A ∩ S ∩ T|) = 125 - (50 + 35 - 20) = 60<br />* **Techno only:** |T| - (|A ∩ T| + |S ∩ T| - |A ∩ S ∩ T|) = 100 - (40 + 35 - 20) = 45<br /><br />**4. Calculate the number of students who prefer at least one brand:**<br /><br />|A ∪ S ∪ T| = |A| + |S| + |T| - |A ∩ S| - |A ∩ T| - |S ∩ T| + |A ∩ S ∩ T|<br /> = 150 + 125 + 100 - 50 - 40 - 35 + 20<br /> = 270<br /><br />**5. Calculate the number of students who prefer none of the three brands:**<br /><br />|U| - |A ∪ S ∪ T| = 300 - 270 = 30<br /><br />**Therefore:**<br /><br />* **Apple only:** 80 students<br />* **Samsung only:** 60 students<br />* **Techno only:** 45 students<br />* **None of the three brands:** 30 students<br />