Chapter 12.5, Problem 1AYU

The ______ ______ is a triangular display of the binomial coefficients.

To determine

To fill: The term that is used to refer a triangular display of the binomial coefficients.

Answer to Problem 1AYU

Solution:

The Pascal triangle is a triangular display of the binomial coefficient

Explanation of Solution

Given:

It is asked to fill in the term of a triangular display of the binomial coefficients.

The symbol $\left(\begin{array}{c}n\\ j\end{array}\right)$ read “$n$ taken $j$ at a time”, is defined

If $j$ and $n$ are integers with $0 \le j \le n$, the symbol $\left(\begin{array}{c}n\\ j\end{array}\right)$ is defined as $\left(\begin{array}{c}n\\ j\end{array}\right) = \frac{n!}{j!\left(n - j\right)!}$

Suppose that the values of the symbol are arranged in a triangular display as shown in the below figure,

This display is called the Pascal Triangle, named after Blaise Pascal a French mathematician.

The Pascal triangle has 1’s down the sides. To get any other entry, add the two nearest entries in the row above it. The shaded triangles in the figure illustrate this feature of the Pascal triangle.