# Show that if there are 30 students in a class, then at least two have same names that begin with the same letter. Explain the pascal’s triangle.

Solution:

Let n = number of students = 30

Let m = number of alphabet = 26

Now,

$$k = \left [ \frac{n}{m} \right ]$$ = $$\left [ \frac{30}{26} \right ]$$ = 1.5 ≈ 2

So, By the Pigeonhole Principle, at least two have same names that begin with same letter.

Pascal’s Triangle

Pascal’s Triangle is a kind of number pattern. Pascal’s Triangle is the triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The numbers are so arranged that they reflect as a triangle. Firstly, 1 is placed at the top, and then we start putting the numbers in a triangular pattern. The numbers which we get in each step are the addition of the above two numbers. It is similar to the concept of triangular numbers.

The nth row in the triangle consists of the bionomial cofficients.

$$\left ( \begin{matrix}n\\ k\end{matrix} \right )$$ 0, k = 0, 1, ……….. n

This triangle is known as Pascal triangle.