91Ó°ÊÓ

The following result is known as Vandermonde's identity, after the German mathematician Abnit-Theophile Vandermonde \((1735-1796) :\) $$ \left(\begin{array}{c}{m+n} \\\ {r}\end{array}\right)=\left(\begin{array}{c}{m} \\\ {0}\end{array}\right)\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)+\left(\begin{array}{c}{m} \\\ {1}\end{array}\right)\left(\begin{array}{c}{n} \\\ {1}\end{array}\right)\left(\begin{array}{c}{n} \\\ {r-1}\end{array}\right)+\left(\begin{array}{c}{m} \\\ {2}\end{array}\right)\left(\begin{array}{c}{n} \\\ {r-2}\end{array}\right)+\cdots+\left(\begin{array}{c}{m} \\\ {r}\end{array}\right)\left(\begin{array}{c}{n} \\ {0}\end{array}\right) $$ Prove Vandermonde's identity algebraically. [Hint: Consider \((1+x)^{m}(x+1)^{n}=(1+x)^{m+n} . ]\)

Prove each. $$C_{n}=\frac{1}{n+1} C(2 n, n), \quad n \geq 0$$

Prove by induction that \(1 \cdot 1 !+2 \cdot 2 !+\cdots+n \cdot n !=(n+1) !-1, n \geq 1\).

The following result is known as Vandermonde's identity, after the German mathematician Abnit-Theophile Vandermonde \((1735-1796) :\) $$ \left(\begin{array}{c}{m+n} \\\ {r}\end{array}\right)=\left(\begin{array}{c}{m} \\\ {0}\end{array}\right)\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)+\left(\begin{array}{c}{m} \\\ {1}\end{array}\right)\left(\begin{array}{c}{n} \\\ {1}\end{array}\right)\left(\begin{array}{c}{n} \\\ {r-1}\end{array}\right)+\left(\begin{array}{c}{m} \\\ {2}\end{array}\right)\left(\begin{array}{c}{n} \\\ {r-2}\end{array}\right)+\cdots+\left(\begin{array}{c}{m} \\\ {r}\end{array}\right)\left(\begin{array}{c}{n} \\ {0}\end{array}\right) $$ Find a formula for \(\sum_{i=2}^{n}\left(\begin{array}{l}{i} \\\ {2}\end{array}\right)\)

Find a formula for \(\sum_{i=2}^{n}\left(\begin{array}{l}i \\\ 2\end{array}\right).\)

Find the number of ternary words that have: Length 3 and are palindromes.

Prove each. $$C_{n}=\frac{2(2 n-1)}{n+1} C_{n-1}, \quad n \geq 1$$

Find the number of ternary words that have: Length 4 and are palindromes.

Find a formula for \(\sum_{i=3}^{n}\left(\begin{array}{l}i \\\ 3\end{array}\right).\)

Show that \((n !) !>(2 n) !,\) if \(n>3\).