ern5231 wrote:In a university there are 30 applicants. These applicants are recommended by three teachers. 15 individuals received recommendation of the first teacher,17 by the second and 20 by the third. What is the least number of applicants who received recommendation from all three teachers?
A) 0 B) 2 C) 3 D) 5 E) 6
Let's solve this with a lot less math and a lot more intuition.
We know that there are 30 total applicants and at least some of them were recommended by more than 1 teacher (since the number of recommendations is greater than 30).
We have 30 students and 15+17+20 = 52 recommendations; so, we need to make up 22 extra recommendations.
We want to minimize the number of triples, so let's see if we can get all 22 people in double recommendations.
Let's say half of the 20 people recommended by the 3rd teacher were also recommended by exactly one of the first two teachers; that takes care of 20/22 duplicates.
Now, we still have 5 singles by the first teacher and 7 singles by the 2nd teacher; we can have 2 of those people recommended by both, accounting for all 22 of our duplicates. Accordingly, we don't need any triplicates at all: choose (A).
To illustrate the above solution, let's call our duplicates A through V:
Teacher 1: ABCDEFGHIJUV + 3 singles (15 students)
Teacher 2: KLMNOPQRSTUV + 5 singles (17 students)
Teacher 3: ABCDEFGHIJKLMNOPQRST (20 students)
