91Ó°ÊÓ

Using the recursive definition of \(c_{n},\) compute each. $$c_{8}$$

Using the binomial theorem, prove each. \(\sum_{r=0}^{n} 2^{r}\left(\begin{array}{l}n \\ r\end{array}\right)=3^{n}\)

A botanist would like to plant three coleus, four zinnias, and five dahlias in a row in her front garden. How many ways can she plant them if: Plants of the same family must be next to each other.

Find the number of solutions to each equation. $$x_{1}+x_{2}+x_{3}+x_{4}=11, x_{1}, x_{2} \geq 2,2 \leq x_{3} \leq 4, x_{4} \geq 3$$

Find the number of ways a committee of five can be formed from a group of five boys and four girls, if each committee must contain: At least two boys.

A botanist would like to plant three coleus, four zinnias, and five dahlias in a row in her front garden. How many ways can she plant them if: The family of zinnias must be in between the other two families.

A die is rolled four times. Find the probability of obtaining: All sixes.

Find the number of solutions to each equation. $$x_{1}+x_{2}+x_{3}+x_{4}=11, x_{1}, x_{2} \geq 2,2 \leq x_{3} \leq 4, x_{4} \geq 3$$

Using the binomial theorem, prove each. \(\sum_{r=0}^{n}\left(\begin{array}{c}n \\\ r\end{array}\right)\left(\begin{array}{c}n \\\ n-r\end{array}\right)\left(\begin{array}{c}2 n \\ n\end{array}\right)\) [Hint: Consider \((1+x)^{2 n}=(1+x)^{n}(1+x)^{n} .\) Equate the coefficients of \(\left.x^{n} \text { from either side. }\right]\)

Find the number of ways seven boys and three girls can be seated in a row if: A boy sits at each end of the row.