Question

What is the row of Pascal's triangle containing the binomial coefficients$\left(\begin{array}{l}9\\ k\end{array}\right)\mathbf{,}\mathbf{0}\mathbf{\le }\mathit{k}\mathbf{\le }\mathbf{9}$?

Short answer

The row of$\left(\begin{array}{l}9\\ k\end{array}\right)$ is then 1 9 36 84 126 126 84 36 9 1.

Step-by-step solution

Use Binomial theorem

Binomial theorem: binomial theorem, statement that for any positive integer n, the nth power of the sum of two numbers x and y may be expressed as the sum of n + 1 terms of the form.

$\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}{\mathbf{)}}^{\mathbf{n}}\mathbf{=}\mathbf{\sum }_{\mathbf{j}\mathbf{=}\mathbf{0}}^{\mathbf{n}}\mathbf{ }\left(\begin{array}{l}n\\ j\end{array}\right){\mathit{x}}^{\mathbf{n}\mathbf{-}\mathbf{j}}{\mathit{y}}^{\mathbf{j}}$

The row of$\left(\begin{array}{l}9\\ k\end{array}\right)$are the binomial coefficients $\left(\begin{array}{l}9\\ k\end{array}\right)$

Evaluate at k=0,1,2,3,4,5,6,7,8,9

$\begin{array}{r}\left(\begin{array}{l}9\\ 0\end{array}\right)=\frac{9!}{0!(9-0)!}\\ =\frac{9!}{0!9!}\\ =1\\ \left(\begin{array}{l}9\\ 1\end{array}\right)=\frac{9!}{1!(9-1)!}\\ =\frac{9!}{1!8!}\\ =9\\ \left(\begin{array}{l}9\\ 2\end{array}\right)=\frac{9!}{2!(9-2)!}\\ =\frac{9!}{2!7!}\\ =36\\ \left(\begin{array}{l}9\\ 3\end{array}\right)=\frac{9!}{3!(9-3)!}\\ =\frac{9!}{3!6!}\\ =84\end{array}$

Similarly:

$\begin{array}{r}\left(\begin{array}{l}9\\ 4\end{array}\right)=\frac{9!}{4!(9-4)!}\\ =\frac{9!}{4!5!}\\ =126\\ \left(\begin{array}{l}9\\ 5\end{array}\right)=\frac{9!}{5!(9-5)!}\\ =\frac{9!}{5!4!}\\ =126\\ \left(\begin{array}{l}9\\ 6\end{array}\right)=\frac{9!}{6!(9-6)!}\\ =\frac{9!}{6!3!}\\ =84\\ \left(\begin{array}{l}9\\ 7\end{array}\right)=\frac{9!}{7!(9-7)!}\\ =\frac{9!}{7!2!}\\ =36\end{array}$

Similarly:

$\begin{array}{r}\left(\begin{array}{l}9\\ 8\end{array}\right)=\frac{9!}{8!(9-8)!}\\ =\frac{9!}{8!1!}\\ =9\\ \left(\begin{array}{l}9\\ 9\end{array}\right)=\frac{9!}{9!(9-9)!}\\ =\frac{9!}{9!0!}\\ =1\end{array}$

Thus, the row of$\left(\begin{array}{l}9\\ k\end{array}\right)$ is then 1 9 36 84 126 126 84 36 9 1.