Harvesting Fruit Toward the middle of the harvesting sea-...

Harvesting Fruit Toward the middle of the harvesting sea- son, peaches for canning come in three types, early, late, and extra late, depending on the expected date of ripening. During a certain week, the following data were recorded at a fruit delivery station: 34 trucks went out carrying early peaches; 61 carried late peaches; 50 carried extra late; 25 carried early and late; 30 carried late and extra late; 8 carried early and extra late; 6 carried all three; 9 carried only figs (no peaches at all). a. How many trucks carried only late variety peaches? b. How many carried only extra late? c. How many carried only one type of peach? d. How many trucks (in all) went out during the week?

Harvesting Fruit Toward the middle of the harvesting sea- son, peaches for canning come in three types, early, late, and extra late, depending on the expected date of ripening. During
a certain week, the following data were recorded at a fruit
delivery station:
34 trucks went out carrying early peaches;
61 carried late peaches;
50 carried extra late;
25 carried early and late;
30 carried late and extra late;
8 carried early and extra late;
6 carried all three;
9 carried only figs (no peaches at all).
a. How many trucks carried only late variety peaches?
b. How many carried only extra late?
c. How many carried only one type of peach?
d. How many trucks (in all) went out during the week?

Key Concepts

Combinatorial Counting

Combinatorial counting involves techniques used to count, arrange, and combine items in a collection, especially when there are overlapping conditions or multiple categories. It is a crucial concept in solving problems where accurate enumeration of exclusive groups is required.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a counting technique used to compute the total number of elements in the union of several sets by systematically adding and subtracting the counts of individual and intersecting sets. This method corrects for any over-counting that occurs when elements belong to multiple sets.

Venn Diagrams

Venn diagrams are visual tools used to represent sets and their relationships through overlapping circles. They help in intuitively understanding how different groups intersect and are particularly useful for solving problems that involve multiple categories with shared elements.

Set Theory

Set theory provides the foundation for grouping objects into collections and understanding their relationships, such as unions, intersections, and differences. It is essential for approaching problems that involve multiple overlapping groups and for organizing data in a structured way.

Transcript

00:06 Given the data of the drugs carrying early late and extra late peaches for canning depending on the expected day per date of ripening so it's given 34 drugs went out carrying early peaches.

00:29 Let us denote early peaches by e that is set of all trucks.

00:38 Carrying early peaches be e so n of e is given as 34 and 61 carried late features let us denote by l n of l is equal to 61 and 50 carry extra late let us denoted by e l that is going to be 50 and 25 carried early and late features, that is n of the intersection l is equal to 25 and 30 carried late on extra late features.

01:51 That is late l intersection extra late is 30 and 8 carried early and extra late.

02:06 E intersection el is equal to 8 and 6 carried all the 3 that is n of e intersection l intersection el is equal to 6 and 9 carried only fixed and no features at all that is e complement intersection l complement intersection el complement is equal to 9 now before proceeding we'll draw the band diagram corresponding to this data now let this set be e let this be l let this be and universal set it's going to be you now given a intersection l intersection el as six so which this part is going to be six and given this is given the intersection l has 25 that is his entire part as 25 now we have to find this particular part which is nothing but e intersection l intersection complement of el that is going to be 25 minus 6 which is 19 and given l intersection el as 30 therefore this area is given as 30 whereas we need just l intersection el intersection complement e that is just we need this area so that is going to be 30 minus 6 which is 24 and given e intersection el as 8 this entire region is 8 now we need just e intersection el intersection complement of l that is going to be 8 minus 6 which is 2 and given number of elements in es 34 so this entire area the exentary region as should have 34 elements but we just need number of elements in the intersection l complement intersection el complement which is going to be this part we can calculate it by the total number of elements in e there is 34 minus is 19 34 minus 19 minus 19 minus 6 minus 2 that is equal to 7 this part is 7 similarly number of elements in l total number of elements in l is 61 this entire region is 61.

07:38 So number of elements in l intersection, e complement intersection, el complement is equal to 61 minus 19 minus 6 minus 24.

08:01 That is equal to 12.

08:29 And given number of elements in els 50.

08:33 So we have to calculate number of elements in elr.

08:39 Section e complement intersection l complement that is going to be the total number of elements in the is el is 50 minus 2 minus 6 minus 24 that is equal to 28 that is equal to 18 use part as 18 and also given number of elements in e complement l complement in el complement will be the elements in the remaining region.

10:01 So that is going to be nine.

10:04 So option a, we have to find the number of trucks carrying only late variety beaches...

Please give Ace some feedback