The 24 colleges of interest to a high school senior include 8 that are expensive (tuition more than $20,000 per year), 8 that are far from home (more than 200 miles away), and 5 that are both expensive and far from home.
A. If the student decides to select a college that is not expensive and within 200 miles of home, how many selections are possible?
B. If the student decides to attend a college that is not expensive and within 200 miles from home during his first two years of college, and then will transfer to a college that is not expensive but is far from home, how many selections of two colleges are possible?

The correct answers are A. 13; B.39.

Step-by-step explanation:

There are 24 colleges (U) that meet the interest of a high school senior.

8 colleges are expensive, 8 are far (above 200 miles from home) and 5 are both expensive and far.

Think of a Venn Diagram where there are two classes namely A = number of expensive colleges and B = number of far colleges. and A ∩ B = 5.

There are 8 - 5 = 3 colleges that are only expensive ( A - B).

There are 8 - 5 = 3 colleges that are only far (200 miles from the house) (B - A).

There are 24 - 8 = 16 colleges that are not expensive (A complement).

There are 24 - 8 = 16 colleges that are not far (B complement).

A. The number of colleges that is not expensive and within 200 miles of home = U - A - (B - A) = 24 - 8 - 3 = 13.

B.The number of colleges that is not expensive and within 200 miles from home during his first two years of college, and then will transfer to a college that is not expensive but is far from home = choices for first type of college × choices for the second type of college = 13 × (A complement ∩ (B - A) = 13 × 3 = 39.

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