Question

A certain college class has 40 students. All the students in the class are known to be from 17 through 34 years of age. You want to make a bet that the class contains at least x students of the same age. How large can you make x and yet be sure to win your bet?​

Answer 1

Answer:

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Answer 2

Given: A certain college class has 40 students. All the students in the class are known to be from 17 through 34 years of age. You want to make a bet that the class contains at least x students of the same age.

To find: How large can you make x and yet be sure to win your bet?​

Solution:

This problem will be solved using the pigeonhole principle which states that for any function f, form a finite set X with n elements to a finite set Y with m elements and for any integer k, if k < n/m, then there exists an element y ∈ Y that y is the image of at least k+1 distinct element of X.

According to this principle,

Let,

A = set of 40 students

B = set of all ages from 17 to 34

Defining a function, f : A⇒B

And, f(x) = age of x , for x is an element of A

Elements contained by A = 40 elements

Elements contained by B = 34 - 17 + 1 = 18 elements

⇒$\frac{A}{B}$ = $\frac{40}{18}$ = 2.2222...

Now, take note of the largest integer k such that k < 40/18 is k = 2. This means that the largest positive integer k for which we can use the above-stated principle is 2.

So, according to the pigeonhole principle,

'y' exists that has an image of k + 1, that is, 2 + 1 = 3.

Thus, you can win the bet if you choose x = 2 + 1 = 3.