SECTION 13.1 Counting 913 Formula (2) can be generalized. THEOREM General Addition Principle of Counting If, for n sets A A A , , , ,n 1 2 … no two have elements in common, then ( ) ( ) ( ) ( ) ∪ ∪ ∪ = + + + n A A A n A n A n A n n 1 2 1 2 (3) Counting Table 1 lists the level of education for all United States residents 25 years of age or older in 2022. EXAMPLE 3 Level of Education Number of U.S. Residents at Least 25Years Old Not a high school graduate 23,825,064 High school graduate 59,908,662 Some college, but no degree 43,842,184 Associate’s degree 20,222,893 Bachelor’s degree 49,641,033 Advanced degree 32,267,301 Source: American Community Survey. Table 1 (a) How many U.S. residents 25 years of age or older had an associate’s degree or a bachelor’s degree? (b) How many U.S. residents 25 years of age or older had an associate’s degree, a bachelor’s degree, or an advanced degree? Solution Let A represent the set of associate’s degree holders, B represent the set of bachelor’s degree holders, and C represent the set of advanced degree holders. No two of the sets A, B, and C have elements in common (although the holder of an advanced degree certainly also holds a bachelor’s degree, the individual would be part of the set for which the highest degree has been conferred). Then ( ) ( ) ( ) = = = n A n B n C 20,222,893 49,641,033 32,267,301 (a) Using formula (2), ( ) ( ) ( ) ∪ = + = + = n A B n A n B 20,222,893 49,641,033 69,863,926 There were 69,863,926 U.S. residents 25 years of age or older who had an associate’s degree or a bachelor’s degree. (b) Using formula (3), ( ) ( ) ( ) ( ) ∪ ∪ = + + = + + = nABC nA nB nC 20,222,893 49,641,033 32,267,301 102,131,227 There were 102,131,227 U.S. residents 25 years of age or older who had an associate’s degree, a bachelor’s degree, or an advanced degree. Now Work PROBLEM 31 Counting the Number of Possible Meals The fixed-price dinner at Mabenka Restaurant provides the following choices: Appetizer: soup or salad Entrée: baked chicken, broiled beef patty, beef liver, or roast beef au jus Dessert: ice cream or cheesecake How many different meals can be ordered? EXAMPLE 4 (continued) 3 Solve Counting Problems Using the Multiplication Principle
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