Question

Prove Pascal’s identity, using the formula for$\left(\begin{array}{l}n\\ r\end{array}\right)$ .

Short answer

Pascal identity,$\left(\begin{array}{c}n+1\\ r\end{array}\right)=\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{l}n\\ r\end{array}\right)$ is proved.

Step-by-step solution

Definition of Pascal’s rule

Inmathematics, Pascal's rule(or Pascal's formula) is acombinatorial identity aboutbinomial coefficients. It states that for positivenatural numbersand,

.$\left(\begin{array}{c}n+1\\ r\end{array}\right)\mathbf{=}\left(\begin{array}{c}n\\ r-1\end{array}\right)\mathbf{+}\left(\begin{array}{l}n\\ r\end{array}\right)$

Where,$\left(\begin{array}{l}n\\ r\end{array}\right)$ Where is a binomial coefficient, one interpretation of which is the coefficient of the$x^r$ term in the expansion of${\left(1+x\right)}^n$ . There is no restriction on the relative sizes of n and r , since, if n < r the value of the binomial coefficient is zero and the identity remains valid.

Use the formula for (nr) and prove the Pascal’s identity

Letbe a positive integer andan integer with$0\le r\le n$

$\begin{array}{r}\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!}{(r-1)!(n-(r-1\left)\right)!}+\frac{n!}{r!(n-r)!}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\\ =\frac{n!\cdot r}{(r-1)!(n-r+1)!\cdot r}+\frac{n!\cdot (n-r+1)}{r!(n-r)!\cdot (n-r+1)}\\ =\frac{n!\cdot r}{r!(n-r+1)!}+\frac{n!\cdot (n-r+1)}{r!(n-r+1)!}\\ =\frac{n!\cdot r+n!\cdot (n-r+1)}{r!(n-r+1)!}\end{array}$

$\begin{array}{r}\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{l}n\\ r\end{array}\right)=\frac{n!\cdot (r+(n-r+1\left)\right)}{r!(n-r+1)!}\\ =\frac{n!\cdot (n+1)}{r!(n+1-r)!}\\ =\frac{(n+1)!}{r!(n+1-r)!}\\ =\left(\begin{array}{c}n+1\\ r\end{array}\right)\end{array}$