A mall has three department stores, J C Penney, Macy’s, and Sears. A survey of 1863 people exiting the mall found that 279 made a purchase at J C Penney, 240 made a purchase at Macy’s, 364 made a purchase at Sears, 76 made a purchase at J C Penney and Macy’s, 108 made a purchase at J C Penney and Sears, 95 of them made a purchase at Macy’s and Sears, and 23 of them made a purchase at all three stores.

a) How many people made a purchase at Macy’s but did not make a purchase at Sears?

b) How many people made a purchase at exactly one of the three stores?

To solve this problem, we will use the principle of inclusion-exclusion.

a) To find the number of people who made a purchase at Macy's but did not make a purchase at Sears, we need to subtract the number of people who made a purchase at both Macy's and Sears from the total number of people who made a purchase at Macy's.

Number of people who made a purchase at Macy's = 240
Number of people who made a purchase at Macy's and Sears = 95

Therefore, the number of people who made a purchase at Macy's but did not make a purchase at Sears is:
240 - 95 = 145

So, 145 people made a purchase at Macy's but did not make a purchase at Sears.

b) To find the number of people who made a purchase at exactly one of the three stores, we need to add the number of people who made a purchase at each store individually and then subtract the number of people who made a purchase at more than one store.

Number of people who made a purchase at J C Penney = 279
Number of people who made a purchase at Macy's = 240
Number of people who made a purchase at Sears = 364
Number of people who made a purchase at all three stores = 23

Therefore, the number of people who made a purchase at exactly one of the three stores is:
(279 + 240 + 364) - (76 + 108 + 95) + (23) = 627 - 279 = 348

So, 348 people made a purchase at exactly one of the three stores.

To answer these questions, we can use a method called the Inclusion-Exclusion Principle.

a) To find the number of people who made a purchase at Macy's but did not make a purchase at Sears, we need to subtract the number of people who made a purchase at both Macy's and Sears from the total number of people who made a purchase at Macy's.

We are given:
- Total number of people who made a purchase at Macy's (240)
- Number of people who made a purchase at both Macy's and Sears (95)

Therefore, the number of people who made a purchase at Macy's but did not make a purchase at Sears is:
240 - 95 = 145.

So, 145 people made a purchase at Macy's but did not make a purchase at Sears.

b) To find the number of people who made a purchase at exactly one of the three stores, we can add up the number of people who made a purchase at each individual store and subtract the overlapping counts.

We are given:
- Number of people who made a purchase at J C Penney (279)
- Number of people who made a purchase at Macy's (240)
- Number of people who made a purchase at Sears (364)
- Number of people who made a purchase at J C Penney and Macy's (76)
- Number of people who made a purchase at J C Penney and Sears (108)
- Number of people who made a purchase at Macy's and Sears (95)
- Number of people who made a purchase at all three stores (23)

To find the number of people who made a purchase at exactly one of the three stores, we use the formula:

(people who made a purchase at J C Penney) + (people who made a purchase at Macy's) + (people who made a purchase at Sears)
- 2 * (people who made a purchase at J C Penney and Macy's)
- 2 * (people who made a purchase at J C Penney and Sears)
- 2 * (people who made a purchase at Macy's and Sears)
+ 3 * (people who made a purchase at all three stores)

Replacing the given values into the formula:
279 + 240 + 364
- 2 * 76
- 2 * 108
- 2 * 95
+ 3 * 23

Simplifying this calculation, we get:
883

Therefore, 883 people made a purchase at exactly one of the three stores.